Pierre Allard
email: piallard@irisa.fr

Sébastien Ferré
email: ferre@irisa.fr

Olivier Ridoux
email: ridoux@irisa.fr

Discovering Functional Dependencies and Association Rules by Navigating in a Lattice of OLAP Views

Keywords: Functional Dependencies, Association Rules, FCA, OLAP, Navigation

Discovering dependencies in data is a well-know problem in database theory. The most common rules are Functional Dependencies (FDs), Conditional Functional Dependencies (CFDs) and Association Rules (ARs). Many tools can display those rules as lists, but those lists are often too long for inspection by users. We propose a new way to display and navigate through those rules. Display is based on On-Line Analytical Processing (OLAP), and organize a set of rules as a cube, where dimensions correspond to the premises of rules. Cubes reflect the hierarchy that exists between FDs, CFDs and ARs. Navigation is based on a lattice, where nodes are OLAP views, and edges are OLAP navigation links, and guides users from cube to cube. We present an illustrative example with the help of our prototype.

Introduction

Discovering dependencies in data is a well-know problem in database theory. Using dependency rules can help to prevent redundancy, to optimize queries and to avoid update errors. There are many softwares for computing dependency rules in a table. They generally provide the rules as a list. The main problem for users is to find the relevant information in those lists. Therefore, users need tools to navigate among them, and to check them. In this paper, we present how to create views over a table, such that users are able to visualize rules. Then we present how to guide users to navigate from a view to another.

We study the discovery of the following kinds of rules. Functional Dependencies (FDs) [START_REF] Codd | The relational model for database management: version 2[END_REF] are dependencies that are valid on entire tables. Conditional Functional Dependencies (CFDs) [START_REF] Fan | Conditional functional dependencies for capturing data inconsistencies[END_REF] are FDs that apply on a subset of a table.

⋆ Pierre Allard benefits from a doctoral grant from Région Bretagne.

Association Rules (ARs) [START_REF] Agrawal | Fast algorithms for mining association rules in large databases[END_REF][START_REF] Zaki | Generating non-redundant association rules[END_REF] are dependencies that apply for particular values of some attributes. An AR can be exact or approximate. Medina and Nourine [START_REF] Medina | A unified hierarchy for functional dependencies, conditional functional dependencies and association rules[END_REF] present the hierarchy that exists between FDs, CFDs and ARs [START_REF] Zaki | Generating non-redundant association rules[END_REF][START_REF] Pasquier | Discovering frequent closed itemsets for association rules[END_REF], with the help of Formal Concept Analysis (FCA) [START_REF] Ganter | Formal Concept Analysis -Mathematical Foundations[END_REF]. Those rules are always in the form premises -conclusion.

The number of rules extracted from a table is often too high. We must therefore provide to the user synthetic views showing a subset of rules. In a number of works, those views are defined by premises and conclusion [START_REF] Chakravarthy | Visualization of association rules over relational DBMs[END_REF]. Most works is about rule visualization, rather than on which view to choose [START_REF] Techapichetvanich | VisAR : A new technique for visualizing mined association rules[END_REF][START_REF] Chakravarthy | Visualization of association rules over relational DBMs[END_REF]. In this paper, we use another database tool, On-Line Analytical Processing (OLAP) to create and navigate between views. OLAP [START_REF] Codd | Provinding OLAP (On-Line Analytical Processing) to User-Analysts: An IT Mandate[END_REF] is often used in Business Intelligence environments. It allows users to aggregate (e.g. sum, average) data at several granularity levels, without knowing a query langage, and to display results in charts. We are interested in OLAP because of the OLAP data structure: cubes, i.e. multidimensional representations of data. A cube is defined by a measure (the values, e.g. the sales) and a set of dimensions (e.g., by month). In this paper, we show that dependency rules can be found visually in a cube, because of the similarity of the form premises -conclusion of the dependency rules and the form dimensions -measure of the cubes. Several papers use lattices along OLAP, often for supporting the precomputation of OLAP cubes [START_REF] Shanmugasundaram | Compressed data cubes for OLAP aggregate query approximation on continuous dimensions[END_REF]. Casali et al. [START_REF] Casali | Extracting semantics from data cubes using cube transversals and closures[END_REF] show a method to organize cubes in a closed cube lattice, to improve the computation of aggregations. Medina and Nourine [START_REF] Medina | Conditional functional dependencies: An FCA point of view[END_REF] create a concept lattice, where each concept is a set of dimensions. Their work allows to discover dependency rules from the lattice.

The number of different cubes can be too high to vizualize them all. Therefore, users need tools to navigate from cube to cube. We relate this work to Logical Information Systems (LIS) [START_REF] Ferré | Logical information systems: from taxonomies to logics[END_REF]. LIS allow users to browse a context (in the sense of FCA) by navigating from concept to concept. The most common OLAP navigation links allow to add or remove a dimension, or to change the granularity level of a dimension. We show that the OLAP navigation links can be used in addition to LIS navigation links.

The main motivation for this paper is to demonstrate that OLAP offers a good support to display and navigate the dependency rules. First we introduce the concepts and definitions, along with illustrative examples, that are needed in our study (Section 2). Next we present how to visualize dependency rules in a cube (Section 3). Then we present the navigation part, to navigate from cube to cube (Section 4). Finally, we detail an example of navigation (Section 5), and conclude (Section 6).

Background and Definitions

A relation, as in databases, is comparable to a many-valued context. A relation schema R is defined by a set of attributes Attr(R). The domain of each attribute A ∈ Attr(R) is denoted by Dom(A). An instance of a relation schema R, a relation r, is a set of transactions. Each transaction t maps a value to each attribute. The notation t[X] represents the values of the transaction t, for the attribute sequence X. Table 2 shows the example relation r e , which is an instance of R e and contains a set of sales. This relation is extracted from [START_REF] Medina | A unified hierarchy for functional dependencies, conditional functional dependencies and association rules[END_REF]. We only changed the labels of values to render the relation less abstract and to add granularity levels (used in Section 3.2).

In this article, we only study dependency rules whose conclusion has a single attribute. A Functional Dependency (FD) [START_REF] Codd | The relational model for database management: version 2[END_REF] expresses the fact that some attribute (the conclusion) of a relation is determined by a set of other attributes (the premises). We only study exact FDs and not Approximate Dependencies [START_REF] Kivinen | Approximate inference of functional dependencies from relations[END_REF].

A FD X → Y , with X ⊆ Attr(R) and

Y ∈ Attr(R), is valid on r if ∀t 1 , t 2 ∈ r, (t 1 [X] = t 2 [X]) ⇒ (t 1 [Y ] = t 2 [Y ]).
A Conditional Functional Dependency (CFDs) [START_REF] Fan | Conditional functional dependencies for capturing data inconsistencies[END_REF][START_REF] Medina | A unified hierarchy for functional dependencies, conditional functional dependencies and association rules[END_REF] is defined by a pair ϕ = (X → Y, T p ), where X → Y is a standard FD, and

T p ⊆ r is a set of patterns called tableau. ϕ is valid if ∀t 1 , t 2 ∈ T p , (t 1 [X] = t 2 [X]) ⇒ (t 1 [Y ] = t 2 [Y ]).
For example, ϕ e = (P roduct → N umber, {( , , , BM X, , ), (01/23, , , , , )}) represents the FD P roduct → N umber restricted to the subset of sales where P roduct = BM X or Date = 01/23, whatever the other attribute values are. The notation represents any value of the corresponding attribute.

An Association Rule (AR) [START_REF] Agrawal | Fast algorithms for mining association rules in large databases[END_REF] expresses the fact that the value of an attribute (the conclusion) is determined by the values of other attributes (the premises).

An AR d is denoted by d = X → Y , where X = ((A 1 = b 1 ) ∧ . . . ∧ (A p = b p ) and Y = (A q = b q )), with A i ∈ Attr(R) and b i ∈ Dom(A i ).
The support of an AR is the number of transactions matching both the premises and the conclusion. The confidence conf (d, r) of d is the ratio of transactions respecting the AR. When conf (d, r) = 1, the AR is said exact. Traditionally, users are interested in exact ARs Approximate ARs (AAR) [START_REF] Zaki | Generating non-redundant association rules[END_REF][START_REF] Pasquier | Discovering frequent closed itemsets for association rules[END_REF] are ARs having a confidence < 1.

Huhtala et al. [START_REF] Huhtala | Tane: An efficient algorithm for discovering functional and approximate dependencies[END_REF] use the notion of X-complete relation [START_REF] Bra | Conditional dependencies for horizontal decompositions[END_REF] and a closure operator to find FDs.

A relation r is X-complete if ∀t 1 , t 2 ∈ r, t 1 [X] = t 2 [X].
A relation r can be decomposed into a set of subsets r ′ ⊆ r, where each r ′ is X-complete. Applying a closure operator, it is possible to form FDs. The Xcomplete partition lattice consists of concepts which represent the partitions, ordered by FD and labelled to find CFD. Medina and Nourine [START_REF] Medina | Conditional functional dependencies: An FCA point of view[END_REF] show the Xcomplete partition lattice of a relation. Each concept shows the set of attributes X and a tableau T p , such that the transactions of T p are X-complete. Each edge (X, Y ) is labelled by a tableau T p , such that r |= (X → Y, T p ). The benefits of this lattice is that it gives a synthetic view of the CFDs and ARs of a relation.

Codd introduced the concept of On-Line Analytical Processing (OLAP) [START_REF] Codd | Provinding OLAP (On-Line Analytical Processing) to User-Analysts: An IT Mandate[END_REF]. Originally, OLAP is designed for Business Intelligence. Indeed, it allows users to aggregate quickly large sets of values, depending on study axes, and to create charts. OLAP users do not need to know a specific language to query the database. Data contained in OLAP warehouses are multidimensionally structured. Each fact of the table contains a measure value (the data that will be aggregated), and dimension values for each study axe. For example, the sale amounts (measure) by store, and by date (dimensions). Each dimension can have several levels of granularity. For example, the date dimension can be expressed by day, month or year. An OLAP table is represented by a cube, reflecting the multidimentional structure. OLAP users can trigger navigation links to navigate from cube to cube.

The main problem of OLAP is that the navigation space is restricted by the starting cube. For instance, OLAP users can neither change the measure (e.g., set the store as measure), nor add a dimension (e.g., add a seller dimension). Therefore, a request to the database administrator is necessary to extract a new starting cube. In [START_REF] Bimonte | Towards a spatial multidimensional model[END_REF], authors present an OLAP model to overcome this limitation. The measure is considered as a special dimension, and can be exchanged with the help of a additional navigation links. In recent years, works on OLAP allowed it to scale to large databases, especially with the help of precomputation [START_REF] Shanmugasundaram | Compressed data cubes for OLAP aggregate query approximation on continuous dimensions[END_REF]. In this paper, we focus on the OLAP concepts of cube structure, granularity levels and navigation links. We do not consider here aspects related to the display of charts, and the precomputation of views. We define cube schemas and cubes.

Definition 1 (Cube Schema, Cube). A cube schema C is defind by a tuple of dimensions, Dim(C) = (A 1 , . . . , A p ), and a measure, M eas(C) = A q . A cube c that is an instance of C is defined by a function f : Dom(A 1 )×. . .×Dom(A p ) → Dom(A q ). This total function allows to access the contents of each cell of the cube.

We study OLAP because of the representation of a cube. Indeed, each cell of a cube represents a transaction or a set of transactions, ordered in a cube structure. The mining of FD implies to check, for any combination of the premise values, if the conclusion value is the same. The cube structure presents all combinations of premise values.

Relation projections

In our mining of dependency rules, we introduced OLAP, because of the similarity between the form (premises -conclusion) of the dependency rules, and the form (dimensions -measure) of the cubes. In order to extract rules from a cube, we need to project the relation into a cube.

Projection of a Relation into a Cube

A cube actually forms a particular view of a relation. The main difference is the distinction between dimensions and the measure, which does not exist in a relation. We use this distinction to form dependency rules. For each transaction of the relation, we place the measure value in the cell of the cube corresponding to the dimension values. The measure values of each cell are usually aggregated. In our study, we retain the measure values as a multiset, because we are interested in dependency rules. Therefore, a cube is defined by a total function f that returns for each cell a multiset of measure values. Thereafter, we consider that an attribute has the same domain in the relation schema and the cube schema.

(c) = (D 1 , . . . , D p ) such that D i ∈ Attr(r) for i ∈ [1, p],
and on the measure M eas(c) = D q ∈ Attr(r). The domain of the dimensions are the same as the domain of their corresponding attributes in the relation. The domain of the measure M eas(D q ) is made of multisets over Dom(D q ). The following function f maps each cell of the cube to a multiset of values from the relation:

f (b 0 , . . . , b p ) = {{b q |t ∈ r, (t[D 1 ] = b 0 ) ∧ . . . ∧ (t[D p ] = b p ) ∧ (t[D q ] = b q )}}.
For example, Table 3.1 shows the projection c e of r e , where the dimensions are Dim(c e ) = (Date) and the measure is M eas(c e ) = Seller. The result of the projection is like a traditional OLAP cube, but data are not aggregated. The advantages are that any kind of value can be studied (e.g. strings), and there is no information loss for rule visualization. Now we show that FDs, CFDs and exact ARs can be visually found in the relation projection (cube), and that the hierarchy between those rules is respected. Indeed, a cell of the cube at coordinates D 1 = d 1 , . . . , D p = d p contains the multiset of values taken by the attribute D q , in all transactions such that

D 1 = d 1 , . . . , D p = d p . The definition of exact ARs D 1 = d 1 ∧ . . . ∧ D p = d p → D q = d q is that for any transaction such that D 1 = d 1 ∧ . . . ∧ D p = d p , then D q = d q .
Then, in this cell, if there is only one value in the multiset (e.g. {{a, a}}), there is an exact AR at this coordinate. If there is more than one value (e.g. {{a, a, b}}), the AR is approximate. Thus, the cube is a synthetic view of all the possible ARs following the dimensions and measure. The confidence is equal to the ratio of elements b q on all the elements on the cell at coordinates (b 1 , . . . , b p ):

conf (d, r) = sup(d, r) f (b 1 , . .

. , b p )

.

There exists a hierarchy between FDs, CFDs and ARs [START_REF] Medina | A unified hierarchy for functional dependencies, conditional functional dependencies and association rules[END_REF][START_REF] Medina | Conditional functional dependencies: An FCA point of view[END_REF]. Indeed, a FD X → Y is equivalent to a CFD (X → Y, T p ) where T p = r (i.e. T p selects all transactions in the relation). Moreover, a CFD is equivalent to a set of ARs. For example, the CFD (Date → Seller, {(01/09, , , , , ), (01/ 23,,,,,)}) is equivalent to the set of ARs {(Date = 01/09 → Seller = John), (Date = 01/23 → Seller = Abby)}. There is a complete hierarchy between FDs, CFDs and ARs. Following the same reasoning as for ARs, it is possible to find CFDs and FDs in a cube. Let c be a cube. If each cell of c contains one or zero element in its multiset, then the FD Dim(c) → M eas(c) is valid. That can be justified by the fact that a FD can be decomposed into a set of ARs, from [START_REF] Medina | A unified hierarchy for functional dependencies, conditional functional dependencies and association rules[END_REF].

Theorem 2 (Functional Dependency in the cube). Let D 1 , . . . , D p → D q be an FD in a relation r, and c be a cube with Dim(c) = (D 1 , . . . , D p ), M eas(c) = D q , f be the projection function of c. This FD is valid in r iff, in c:

∀(b 1 , . . . , b p ) ∈ (Dom(D 1 ) × . . . × Dom(D p )), ∀v 1 , v 2 ∈ f (b 1 , . . . , b p ), v 1 = v 2 .
The definition of a CFD in the cube is between the definition of an AR and a FD. If each cell in a subset of the cube (e.g. a line or a square subset) contains one or zero element in its multiset, then there is a valid CFD, whose pattern tableau covers the subset of the cube. The hierarchy established by [START_REF] Medina | A unified hierarchy for functional dependencies, conditional functional dependencies and association rules[END_REF] is here confirmed, according to the number of cells with one or zero element.

For example, Table 3.1 shows dependency rules. We note that each cell except {{Abby, Jim}} has one distinct value. This implies that there are ARs at those cells (e.g. Date = 01/06 → Seller = Abby), and a CFD whose pattern covers those cells ((Date → Seller, {(01/09, , , , , ), . . .})). On the contrary, no FD can be found from this view.

Granularity Levels

A major strength of OLAP is the definition of taxonomies on the dimensions. Indeed, the dimension values are hierarchically organized, according to several levels of granularity. In traditional OLAP, an aggregation function can synthesize the grouped data, using a function (e.g., sum, average, count). Han and Fu [START_REF] Han | Discovery of multiple-level association rules from large databases[END_REF] show methods to find ARs at several levels of granularity. It is interesting to see how ARs, then FDs, can be seen in a cube with taxonomies. We now formalize the taxonomy of dimensions.

Definition 3 (Dimension taxonomy).

Let D be a dimension. This dimension, at granularity level δ, is denoted by D ∼ δ. A dimension taxonomy on D is defined by a set of n ordered levels ∆ = {1, . . . , n} and a set of rules b i ⊑ b j , with b i ∈ Dom(D ∼ i), b j ∈ Dom(D ∼ j), and j = i -1. The levels form a partition of the dimension values, i.e., Dom(D) = ∪ i∈n Dom(D ∼ i) and ∀i, j ∈ 1 . . . n, Dom(D ∼ i) ∩ Dom(D ∼ j) = ∅. To get a more concise notation, a dimension without taxonomy keeps its old notation. Figure 3.2 presents a dimension taxonomy for the P roduct and Store dimensions. A product can be represented by its type, and a store by its region. The definition of a cube schema is not modified. We redefine a projection function from a relation to a cube, using taxonomies. Definition 4 (Projection of a relation in a cube with taxonomies). Let r be a relation with attributes Attr(r). A cube c is the projection of the relation with taxonomies, on dimensions Dim(c) = (D 1 , . . . , D p ) such that D i ∈ Attr(r), and a set of levels (δ 1 , . . . , δ p ) for each dimension, and a measure M eas(c) = D q , with D q ∈ Attr(r) and δ q a level of D q . The domains of elements of Dim(c) are equivalent to the domains of the corresponding elements of Dom(D i ∼ δ i ). The domain of M eas(c) is equivalent to the multiset of Dom(D q ∼ δ q ). The following function f maps each cell of the cube to a multiset of values from the relation:

f (b ′ 1 , . . . , b ′ p ) = {{b ′ q |t ∈ r, ∀k ∈ 1 . . . p, q, (t[D k ] = b k ∧ b k ⊑ b ′ k )}}.
This definition allows users to create new cubes, at several granularity levels. This implies that more dependency rules can be extracted from the relation. For instance, Table 3.2 shows the result of the projection of r e in c et , with the dimensions Dim(c et ) = (Date ∼ δ month , Seller) and the measure M eas(c et ) = N umber. We can then deduce that the CFD ϕ 2 = (Date ∼ δ month ∧ Seller → N umber, {( , John, , , , ), ( , Jim, , , , )}) is valid on r e .

Navigation

In the previous section we show that a projection of a relation into a cube allows to see the FDs, CFDs and ARs, following the dimensions and the measure of the cube. To give access to all dependency rules, we need to give access to cubes for all dimension combinations, granularity levels, and measures.

To navigate from cube to cube, changing granularity levels, adding or removing dimensions, we use OLAP navigation links. Traditionally, OLAP systems do not have an add navigation link, to add a dimension (because of the initial cube problem). We add this navigation link. The set of navigation links is: Roll-up (dimension / measure) Traditionnaly, a roll-up changes the granularity level of a dimension. Our choice is to consider the measure as a special dimension, then a roll-up can here be used on the measure. Formally, a roll-up on D ∼ δ i will modify it into D ∼ (δ i-1 ). Drill-down (dimension / measure) Such as roll-up, drill-down changes the granularity level of a dimension or a measure. A drill-down on D ∼ δ i will modify it into D ∼ (δ i+1 ). Delete Deletes a dimension. Delete a dimension is a special case of roll-up (at the top level). Add Add a dimension.

Those six navigation links change the number of values in the chosen dimension, and hence the visible dependency rules. A roll-up (resp. drill-down) on a dimension increases (resp. decreases) the number of values in this dimension. This means that the multisets of measure values are splitted (resp. regrouped), which promotes the appearance (resp. disappearance) of rules. Figure 4 shows the set of accessible navigation links. The effect of each navigation link is given for two starting cubes: one verifying an FD, and another not verifying an FD. We detail the two phenomena explained above.

The first phenomena is the splitting of the measure values (e.g. when the user adds a dimension or drill-down a dimension). This splitting is important in the mining of dependency rules. Indeed, we see that when values are dispatched into separate cells, they have less chance to be with different values, then there is a greater chance to have an AR or a FD. There is a similar phenomenon when mining of rules directly from relations. However, one has to be careful neither to over-increase the granularity of the dimensions, nor to over-reduce the granularity of the measure. Indeed, rules have more chance to appear, but their quality and precision decrease.

The second phenomena is the regroupment of the measure values (e.g. when the user deletes a dimension or roll-up a dimension). If equal values are grouped one per product. Each case except ATVs contains one number value. We select all transactions such that the product is not ATV (3). All cells of the resulting cube contains one number value. Therefore, there is a CFD (P roduct → N umber, T p ), where T p contains cells where the product is not ATV. Now we want to work with the store locations. We change the measure to Store (4) then we select all transactions (5). The resulting cube does not contain a FD. Figure 4 shows that adding a dimension helps to have more dependencies. We then add N umber as a new dimension [START_REF] Codd | Provinding OLAP (On-Line Analytical Processing) to User-Analysts: An IT Mandate[END_REF]. The resulting cube has two dimensions and shows the CFD (P roduct, N umber → Store, T p ) with T p containing all the transactions except those where N umber = 3 and P roduct = BM X. Indeed, this cell contains {{Quimper, Brest}}. Those two cities being in the same region, we change the granularity level of the measure to Store ∼ δ Region [START_REF] Codd | The relational model for database management: version 2[END_REF]. The resulting cube shows a FD, because each cell contains zero or one region. Figure 4 shows two possibilities of a roll-up in a dimension, with a starting cube containing FD: either the FD is still valid, or it is made invalid. We roll-up the P roduct dimension to P roduct ∼ δ T ype [START_REF] Fan | Conditional functional dependencies for capturing data inconsistencies[END_REF]. The FD is still valid. Finally, we see that in each column of the current cube, there is only one region. Then we delete the dimension N umber [START_REF] Ferré | Logical information systems: from taxonomies to logics[END_REF]. The final cube shows a FD P roduct ∼ δ T ype → Store ∼ δ Region .

Conclusion

In this paper, we show that projecting a relation into a cube brings relevant properties. First, Association Rules, Conditional Functional Dependencies and Functional Dependencies are made visible in this synthetic view, through the number of values in each cell. This is due to the similarity between the form premises -conclusion of the dependency rules, and the form dimensions -measure of the rules. Then, we show that the hierarchy established in [START_REF] Medina | A unified hierarchy for functional dependencies, conditional functional dependencies and association rules[END_REF] is consistent with our approach, and related to the number of cells containing one or zero values. Using OLAP implies that we can now see the rules at several levels of granularity.

A cube is a representation of a subset of all rules that can be extracted from a relation. We use the conventional OLAP navigation links to allow users to navigate from cube to cube, to add or to remove dimensions, or to change the granularity levels. This paper shows how to guide the user to choose navigation links. The navigation links show that some operators have a predictable behavior about the appearance or disappearance of rules.

Definition 2 (

 2 Projection of a relation into a cube). Let r be a relation with attributes Attr(r). A cube c is obtained by the projection of the relation r on dimensions Dim

Theorem 1 (

 1 Association Rule in the cube). Let d = (D 1 = b 1 )∧. . .∧(D p = b p ) → (D q = b q ) be an AR, and c be a cube with Dim(c) = (D 1 , . . . , D p ), M eas(c) = D q , and f be the projection function of c. The support of this rule is equal to the number of elements that have value b q in the cell of c at coordinates (b 1 , . . . , b p ): sup(d, r) = {{b|b ∈ f (b 1 , . . . , b p ) ∧ b = b q }} .

Fig. 1 .Table 4 .

 14 Fig. 1. The dimension taxonomies of P roduct and Store. Date ∼ δ month 10/01 10/02 Seller John {{2}} {{20}} Abby {{1, 3, 3, 7}} Jim {{4}} {{5}} Bob {{4}} {{6}}Table 4. The projection cet of re, with the dimensions Dim(cet) = (Date ∼ δ month , Seller) and the measure M eas(c) = N umber.

Fig. 2 .

 2 Fig. 2. The several OLAP navigation links. A cell with one or zero point means an exact AR. A cell with two points mean an approximate AR. The bold cubes mean the FD is true.

Fig. 3 .

 3 Fig. 3. The prototype Abilis, at the step 8 of the scenario.

Table 2 .

 2 Abby}} {{Abby, Jim}} {{Bob}} {{John}} {{Jim}} {{Bob}} The projection ce of re, with Dim(ce) = (Date) and M eas(ce) = Seller.

			Date
	01/09	01/16	01/23	02/06	02/13	02/20	02/27
	{{John}} {{Abby,

Table 3 .

 3 The projection ce of re, with Dim(ce) = (Seller, Date) and M eas(ce) = P roduct.Sebastien¿ Il nous fallait trouver un cube a deux dimensions pour que ca soit plus concret. J'ai donc cherche un cube a deux dimensions ou je pouvais apercevoir une cellule avec 2 valeurs egales ({{Abby, Abby}}) et une cellule avec 2 valeurs differentes ({Abby, Bob}}). J'ai cherche avec toutes les combinaisons possibles, et je n'ai pas trouve. Il faudra donc choisir entre un exemple sur un cube digne de ce nom et un cube ou on voit les multi-ensembles... Pierre¿ et si tu aggreges par mois ?

		Seller
	John	Abby	Jim	Bob
	01/09 {{AT V }}
	01/16	{{AT V, BM X}}
	01/23	{{BM X, P ants}} {{Shoes}}
	Date 02/06			{{Shoes}}
	02/13 {{Balloon}}
	02/20	{{Skates}}
	02/27			{{Sneakers}}

Step into a same cell, this does not change the rule at this coordinate. On the contrary, if different values are grouped into a same cell, this breaks the dependency rule at this coordinate. Figure 4 shows the two cases of value regroupments. This figure helps the user to choose navigation links, to find new rules with the help of OLAP navigation links.

Navigation Scenario

In this section, we present our prototype Abilis and we show its use with the example r e . Abilis is a web application based on Logical Information System (LIS) [START_REF] Ferré | Logical information systems: from taxonomies to logics[END_REF]. LIS allows users to query and navigate from concept to concept. LIS defines navigation links to refine the set of objects by navigating to other concepts. We have added a new view of the selected objects as a cube, as well as the OLAP navigation links. We navigate with the example r e , adding a taxonomy on each dimension. Dates are organized by day and month; stores are organized by city and region; product are organized by article and by type. We search for dependency rules between products, numbers and stores. Table 5 presents the steps of our navigation scenario. Figure 5 shows our prototype at step 8. It contains (1) the current query (All means all the transactions), (2) the navigation tree (to add a dimension or refine the query) and (3) the view, displayed as a cube. Each cell of the cube shows the multiset of measure values as a tag cloud. For example, [START_REF] Casali | Extracting semantics from data cubes using cube transversals and closures[END_REF] shows that the support of the rule P roduct ∼ δ T ype = Bike ∧ N umber = 3 → Store ∼ δ Region is 2. The size of an item in a cell depends on the support of the rule.

The first view presents a single cell containing the 10 transactions (0). We set N umber as the measure [START_REF] Agrawal | Fast algorithms for mining association rules in large databases[END_REF]. The cube has always one cell, and displays the proportion of the seven different number values (e.g. there is two sales with N umber = 4). We add a dimension P roduct (2), to check dependencies like P roduct → N umber. The resulting cube partitions the numbers into six cells,


Hagen Neidhardt

Valentin A Zagrebnov

Solution Operator for Non-Autonomous Perturbation of Gibbs Semigroup To the memory of

The paper is devoted to a linear dynamics for non-autonomous perturbation of the Gibbs semigroup on a separable Hilbert space. It is shown that evolution family {U(t, s)} 0≤s≤t solving the non-autonomous Cauchy problem can be approximated in the trace-norm topology by product formulae. The rate of convergence of product formulae approximants {U n (t, s)} {0≤s≤t,n≥1} to the solution operator {U(t, s)} {0≤s≤t} is also established.

Introduction and main result

The aim of the paper is two-fold. Firstly, we study a linear dynamics, which is a non-autonomous perturbation of Gibbs semigroup. Secondly, we prove product formulae approximations of the corresponding to this dynamics solution operator {U(t, s)} {0≤s≤t} , known also as evolution family, fundamental solution, or propagator, see [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF] Ch.VI, Sec.9.

To this end we consider on separable Hilbert space H a linear non-autonomous dynamics given by evolution equation of the type:

∂ u(t) ∂t = -C(t)u(t), u(s) = u s , s ∈ [0, T ) ⊂ R + 0 , C(t) := A + B(t), u s ∈ H, t ∈ I := [0, T ], (1.1)
where R + 0 = {0} ∪R + and linear operator A is generator of a Gibbs semigroup. Note that for the autonomous Cauchy problem (ACP) (1.1), when B(t) = B, the outlined 1 programme corresponds to the Trotter product formula approximation of the Gibbs semigroup generated by a closure of operator A + B, [START_REF] Zagrebnov | Gibbs Semigroups, Operator Theory Series: Advances and Applications[END_REF] Ch. [START_REF] Kato | Abstract evolution equation of parabolic type in Banach and Hilbert spaces[END_REF].

The main result of the present paper concerns the non-autonomous Cauchy problem (nACP) (1.1) under the following Assumptions:

(A1) The operator A ≥ 1 in a separable Hilbert space H is self-adjoint. The family {B(t)} t∈I of non-negative self-adjoint operators in H is such that the bounded operator-valued function (1 + B(•)) -1 : I -→ L(H) is strongly measurable.

(A2) There exists α ∈ [0, 1) such that inclusion: dom(A α ) ⊆ dom(B(t)), holds for a.e. t ∈ I. Moreover, the function B(•)A -α : I -→ L(H) is strongly measurable and essentially bounded in the operator norm:

C α := ess sup t∈I B(t)A -α < ∞. (1.2) (A3) The map A -α B(•)A -α : I -→ L(H)
is Hölder continuous in the operator norm: for some β ∈ (0, 1] there is a constant L α,β > 0 such that one has estimate [START_REF] Ichinose | Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations[END_REF] to prove the operatornorm convergence of product formula approximants {U n (t, s)} 0≤s≤t to solution operator {U(t, s)} 0≤s≤t . Then they were widely used for product formula approximations in [START_REF] Neidhardt | On convergence rate estimates for approximations of solution operators for linear non-autonomous evolution equations[END_REF]- [START_REF] Nickel | Evolution semigroups and product formulas for nonautonomous Cauchy problems[END_REF] in the context of the evolution semigroup approach to the nACP, see [START_REF] Monniaux | Semigroup method to solve non-autonomous evolution equations[END_REF]- [START_REF] Neidhardt | On abstract linear evolution equations[END_REF].

A -α (B(t) -B(s))A -α ≤ L α,β |t -s| β , (t, s) ∈ I × I. (1.3) (A4) The operator A is generator of the Gibbs semigroup {G(t) = e -tA } t≥0 , that is, a strongly continuous semigroup such that G(t)| t>0 ∈ C 1 (H). Here C 1 (H) denotes the * -ideal of trace-class operators in C * -algebra L(H) of bounded operators on H. Remark 1.1 Assumptions (A1)-(A3) are introduced in

Remark 1.2

The following main facts were established (see, e.g., [START_REF] Ichinose | Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations[END_REF][START_REF] Nagel | Well-poseness of nonautonomous abstract Cauchy problems[END_REF][START_REF] Vuillermot | A general TrotterKato formula for a class of evolution operators[END_REF][START_REF] Yagi | Parabolic evolution equation in which the coefficients are the generators of infinitly differentiable semigroups[END_REF]) about the nACP for perturbed evolution equation of the type (1.1): (a) By assumptions (A1)-(A2) the operators {C(t) = A + B(t)} t∈I have a common dom(C(t)) = dom(A) and they are generators of contraction holomorphic semigroups. Hence, the nACP (1.1) is of parabolic type [START_REF] Kato | Abstract evolution equation of parabolic type in Banach and Hilbert spaces[END_REF][START_REF] Sobolevskii | Parabolic equations in a Banach space with an unbounded variable operator, a fractional power of which has a constant domain of definition[END_REF]. (b) Since domains dom(C(t)) = dom(A), t ≥ 0, are dense, the nACP is well-posed with time-independent regularity subspace dom(A). (c) Assumptions (A1)-(A3) provide the existence of evolution family solving nACP (1.1) which we call the solution operator. It is a strongly continuous, uniformly bounded family of operators {U(t, s)} (t,s)∈∆ , ∆ := {(t, s) ∈ I × I : 0 ≤ s ≤ t ≤ T }, such that the conditions U(t,t) = 1 for t ∈ I, U(t, r)U(r, s) = U(t, s), for, t, r, s ∈ I for s ≤ r ≤ t, (1.4) are satisfied and u(t) = U(t, s) u s for any u s ∈ H s is in a certain sense (e.g., classical, strict, mild) solution of the nACP (1.1).

(d) Here H s ⊆ H is an appropriate regularity subspace of initial data. Assumptions (A1)-(A3) provide H s = dom(A) and U(t, s)H ⊆ dom(A) for t > s.

In the present paper we essentially focus on convergence of the product approximants {U n (t, s)} (t,s)∈∆ ,n≥1 to solution operator {U(t, s)} (t,s)∈∆ . Let

s = t 1 < t 2 < . . . < t n-1 < t n < t, t k := s + (k -1) t-s n , (1.5)
for k ∈ {1, 2, . . ., n}, n ∈ N, be partition of the interval [s,t]. Then corresponding approximants may be defined as follows:

W (n) k (t, s) :=e -t-s n A e -t-s n B(t k ) , k = 1, 2, . . . , n, U n (t, s) :=W (n) n (t, s)W (n) n-1 (t, s) × • • • × W (n) 2 (t, s)W (n)
1 (t, s).

(1.6)

It turns out that if the assumptions (A1)-(A3), adapted to a Banach space X, are satisfied for α ∈ (0, 1), β ∈ (0, 1) and in addition the condition α < β holds, then solution operator {U(t, s)} (t,s)∈∆ admits the operator-norm approximation ess sup

(t,s)∈∆ U n (t, s) -U(t, s) ≤ R β ,α n β -α , n ∈ N, (1.7)
for some constant R β ,α > 0. This result shows that convergence of the approximants {U n (t, s)} (t,s)∈∆ ,n≥1 is determined by the smoothness of the perturbation B(•) in (A3) and by the parameter of inclusion in (A2), see [START_REF] Neidhardt | Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems[END_REF].

The Lipschitz case β = 1 was considered for X in [START_REF] Neidhardt | On convergence rate estimates for approximations of solution operators for linear non-autonomous evolution equations[END_REF]. There it was shown that if α ∈ (1/2, 1), then one gets estimate

ess sup t∈I U n (t, s) -U(t, s) ≤ R 1,α n 1-α , n = 2, 3, . . . . (1.8)
For the Lipschitz case in a Hilbert space H the assumptions (A1)-(A3) yield a stronger result [START_REF] Ichinose | Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations[END_REF]:

ess sup (t,s)∈∆ U n (t, s) -U(t, s) ≤ R log(n) n , n = 2, 3, . . . . (1.9)
Note that actually it is the best of known estimates for operator-norm rates of convergence under conditions (A1)-(A3).

The estimate (1.7) was improved in [START_REF] Neidhardt | Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces. Analysis and Operator Theory[END_REF] for α ∈ (1/2, 1) in a Hilbert space using the evolution semigroup approach [START_REF] Evans | Time dependent perturbations and scattering of strongly continuous groups on Banach spaces[END_REF][START_REF] Howland | Stationary scattering theory for time-dependent Hamiltonians[END_REF][START_REF] Neidhardt | On abstract linear evolution equations[END_REF]. This approach is quite different from technique used for (1.9) in [START_REF] Ichinose | Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations[END_REF], but it is the same as that employed in [START_REF] Neidhardt | On convergence rate estimates for approximations of solution operators for linear non-autonomous evolution equations[END_REF].

Proposition 1.3 [12] Let assumptions (A1)-(A3) be satisfied for β ∈ (0, 1). If β > 2α -1 > 0, then estimate ess sup (t,s)∈∆ U n (t, s) -U(t, s) ≤ R β n β , (1.10)
holds for n ∈ N and for some constant R β > 0.

Note that the condition β > 2α -1 is weaker than β > α (1.7), but it does not cover the Lipschitz case (1.8) because of condition β < 1.

The main result of the present paper is the lifting of any known operator-norm bounds (1.7)-(1.10) (we denote them by R α,β ε α,β (n)) to estimate in the trace-norm topology • 1 . This is a subtle matter even for ACP, see [START_REF] Zagrebnov | Gibbs Semigroups, Operator Theory Series: Advances and Applications[END_REF] Ch.5.4.

-The first step is the construction for nACP (1.1) a trace-norm continuous solution operator {U(t, s)} (t,s)∈∆ , see Theorem 2.2 and Corollary 2.3.

-Then in Section 3 for assumptions (A1)-(A4) we prove (Theorem 1.4) the corre-

sponding trace-norm estimate R α,β (t, s)ε α,β (n) for difference U n (t, s) -U(t, s) 1 .
Theorem 1.4 Let assumptions (A1)-(A4) be satisfied. Then the estimate

U n (t, s) -U(t, s) 1 ≤ R α,β (t, s)ε α,β (n) , (1.11)
holds for n ∈ N and 0 ≤ s < t ≤ T for some R α,β (t, s) > 0.

Preliminaries

Besides Remark 1.2(a)-(d) we also recall the following assertion, see, e.g., [START_REF] Sobolevskii | Parabolic equations in a Banach space with an unbounded variable operator, a fractional power of which has a constant domain of definition[END_REF] Note that solution of (1.

1) is called classical if u(t) ∈ C([0, T ], H) ∩ C 1 ([0, T ], H), u(t) ∈ dom(C(t)), u(s) = u s , and C(t)u(t) ∈ C([0, T ],
H) for all t ≥ s, with convention that (∂ t u)(s) is the right-derivative, see, e.g., [START_REF] Sobolevskii | Parabolic equations in a Banach space with an unbounded variable operator, a fractional power of which has a constant domain of definition[END_REF], Theorem 1, or [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF], Ch.VI.9.

Since the involved into (A1), (A2) operators are non-negative and self-adjoint, equation (2.1) implies that the solution operator consists of contractions:

∂ t U(t, s)u 2 = -2(C(t)U(t, s)u,U(t, s)u) ≤ 0, for u ∈ H. (2.2)
By (A1) G(t) = e -tA : H → dom(A). Applying to (2.1) the variation of constants argument we obtain for U(t, s) the integral equation :

U(t, s) = G(t -s) - t s dτ G(t -τ) B(τ)U(τ, s), U(s, s) = 1 .
(2.3)

Hence evolution family {U(t, s)} (t,s)∈∆ , which is defined by equation (2.3), can be considered as a mild solution of nACP (2.1) for 0 ≤ s ≤ t ≤ T in the Banach space L(H) of bounded operators, cf. [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF], Ch.VI.7. Note that assumptions (A1)-(A4) yield for 0 ≤ s < t ≤ T , τ ∈ (s,t) and for the closure A -α B(τ):

G(t -s)A α 1 ≤ M α (t -τ) α and A -α B(τ) ≤ C α .
(2.4)

Then (2.
2), (2.4) give the trace-norm estimate

t s dτ G(t -τ) B(τ)U(τ, s) 1 ≤ M α C α 1 -α (t -s) 1-α , (2.5
S 0 (t, s) = U A (t -s), S n (t, s) = - t s ds G(t -τ) B(τ) S n-1 (τ, s), n ≥ 1.
(2.6)

Since in (2.6) the operators S n≥1 (t, s) are the n-fold trace-norm convergent Bochner integrals

S n (t, s) = t s dτ 1 τ 1 s dτ 2 . . . τ n-1 s dτ n G(t -τ 1 )(-B(τ 1 ))G(τ 1 -τ 2 ) • • • G(τ n-1 -τ n )(-B(τ n ))G(τ n -s), (2.7)
by contraction property (2.2) and by estimate (2.5) there exit 0 ≤ s ≤ t such that M α C α (t -s) 1-α /(1 -α) =: ξ < 1 and

S n (t, s) 1 ≤ ξ n , n ≥ 1. (2.8)
Consequently ∑ ∞ n=0 S n (t, s) converges for t > s in the trace-norm and satisfies the integral equation (2.3). Thus we get for solution operator of nACP the representation

U(t, s) = ∞ ∑ n=0 S n (t, s) .
(2.9)

This result can be extended to any 0 ≤ s < t ≤ T using (1.4). We note that for s ≤ t the above arguments yield the proof of assertions in the next Theorem 2.2 and Corollary 2.3, but only in the strong ([19] Proposition 3.1, main Theorem in [START_REF] Vuillermot | A general TrotterKato formula for a class of evolution operators[END_REF]) and in the operator-norm topology, [START_REF] Ichinose | Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations[END_REF]

(H)) ∩ C 1 ((s, T ], C 1 (H)) with U(s, s) = 1 and U(t, s) ∈ C 1 (H), C(t)U(t, s) ∈ C 1 (H) for t > s, which means that solution U(t, s) of (2.10) is strict, cf. [19] Definition 1.1.
We note that these results for ACP in Banach space C 1 (H) are well-known for Gibbs semigroups, see [START_REF] Zagrebnov | Gibbs Semigroups, Operator Theory Series: Advances and Applications[END_REF], Chapter 4. Now, to proceed with the proof of Theorem 1.4 about trace-norm convergence of the solution operator approximants (1.6) we need the following preparatory lemma. Lemma 2.4 Let self-adjoint positive operator A be such that e -tA ∈ C 1 (H) for t > 0, and let V 1 ,V 2 , . . . ,V n be bounded operators L(H).

Then n ∏ j=1 V j e -t j A 1 ≤ n ∏ j=1 V j e -(t 1 +t 2 +...+t n )A/4 1 , (2.11)
for any set {t 1 ,t 2 , . . . ,t n } of positive numbers.

Proof. At first we prove this assertion for compact operators: V j ∈ C ∞ (H), j = 1, 2, . . . , n. Let t m := min{t j } n j=1 > 0 and T := ∑ n j=1 t j > 0. For any 1 ≤ j ≤ n, we define an integer ℓ j ∈ N by

2 ℓ j t m ≤ t j ≤ 2 ℓ j +1 t m . Then we get ∑ n j=1 2 ℓ j t m > T /2 and n ∏ j=1 V j e -t j A = n ∏ j=1
V j e -(t j -2 ℓ j t m )A (e -t m A ) ℓ j .

(2.12)

By the definition of the • 1 -norm and by inequalities for singular values

{s k (•)} k≥1 of compact operators n ∏ j=1 V j e -t j A 1 = ∞ ∑ k=1 s k n ∏ j=1 V j e -(t j -2 ℓ j t m )A (e -t m A ) 2 ℓ j ≤ ∞ ∑ k=1 n ∏ j=1 s k e -(t j -2 ℓ j t m )A s k (e -t m a ) 2 ℓ j s k (V j ) ≤ ∞ ∑ k=1 s k (e -t m A ) ∑ n j=1 2 ℓ j n ∏ j=1 V j . (2.13)
Here we used that s k (e -(t j -2 ℓ j t m )A ) ≤ e -(t j -2 ℓ j t m )A ≤ 1 and that

s k (V j ) ≤ V j . Let N := ∑ n j=1 2 ℓ j and T m := Nt m > T /2. Since ∞ ∑ k=1
s k (e -tA/q ) q = ( e -tA/q q ) q , the inequality (2.13) yields for q = N:

n ∏ j=1 V j e -t j A 1 ≤ e -T m A/N N N n ∏ j=1 V j . (2.14)
Now we consider an integer p ∈ N such that 2 p ≤ N < 2 p+1 . It then follows that T /4 < T m /2 < 2 p T m /N, and hence we obtain

e -T m A/N q=N N = ∞ ∑ k=1 s N k (e -T m A/N ) (2.15) ≤ ∞ ∑ k=1 s 2 p k (e -2 p T m A/2 p N ) ≤ ∞ ∑ k=1 s 2 p k (e -TA/2 p+2 ) = e -TA/2 2 1 ,
where we used that s k (e -T m A/N ) = s k (e -2 p T m A/2 p N ) ≤ e -T m A/N ≤ 1, and that s k (e -(t+τ)A ) ≤ e -tA s k (e -τA ) ≤ s k (e -τA ) for any t, τ > 0. Therefore, the estimates (2.14), (2.15) give the bound (2.11). Now, let V j ∈ L(H), j = 1, 2, . . . , n, and set Ṽj := V j e -εA for 0 < ε < t m . Hence, Ṽj ∈ C 1 (H) ⊂ C ∞ (H) and s k ( Ṽj ) ≤ Ṽj ≤ V j . If we set t j := t j -ε,

then n ∏ j=1 V j e -t j A 1 ≤ n ∏ j=1 V j e -(t 1 +t 2 +•••+t n )A/4
1 .

(2.16)

Since the semigroup {e -tA } t≥0 is • 1 -continuous for t > 0, we can take in (2.16) the limit ε ↓ 0. This gives the result (2.11) in general case.

3 Proof of Theorem 1.4

We follow the line of reasoning of the lifting lemma developed in [START_REF] Zagrebnov | Gibbs Semigroups, Operator Theory Series: Advances and Applications[END_REF], Ch.5.4.1. 1. By virtue of (1.4) and (1.6) we obtain for difference in (1.11) formula:

U n (t, s) -U(t, s) = 1 ∏ k=n W (n) k (t, s) - 1 ∏ l=n U(t l+1 ,t l ). (3.1)
Let integer k n ∈ (1, n). Then (3.1) yields the representation:

U n (t, s) -U(t, s) = k n +1 ∏ k=n W (n) k (t, s) - k n +1 ∏ l=n U(t l+1 ,t l ) 1 ∏ k=k n W (n) k (t, s) + k n +1 ∏ l=n U(t l+1 ,t l ) 1 ∏ k=k n W (n) k (t, s) - 1 ∏ l=k n U(t l+1 ,t l ) ,
which implies the trace-norm estimate

U n (t, s) -U(t, s) 1 ≤ k n +1 ∏ k=n W (n) k (t, s) - k n +1 ∏ l=n U(t l+1 ,t l ) 1 ∏ k=k n W (n) k (t, s) 1 + k n +1 ∏ l=n U(t l+1 ,t l ) 1 1 ∏ k=k n W (n) k (t, s) - 1 ∏ l=k n U(t l+1 ,t l ) . (3.2)
2. Now we assume that lim n→∞ k n /n = 1/2. Then (1.5) yields lim n→∞ t k n = (t + s)/2, lim n→∞ t n = t and uniform estimates (1.7)-(1.10) with the bound R α,β ε α,β (n) imply ess sup

(t,s)∈∆ k n +1 ∏ k=n W (n) k (t, s) -U(t, (t + s)/2) ≤ R (1) α,β ε α,β (n), (3.3)
ess sup

(t,s)∈∆ 1 ∏ k=k n W (n) k (t, s) -U((t + s)/2, s) ≤ R (2) α,β ε α,β (n), (3.4)
for n ∈ N and for some constants R

{V k = e -t-s n B(t k ) } n k=1 there exists a 1 > 0 such that 1 ∏ k=k n W (n) k (t, s) 1 = 1 ∏ k=k n e -t-s n A e -t-s n B(t k ) 1 ≤ a 1 e -t-s 2 A 1 . (3.5)
Similarly there is a 2 > 0 such that (3.9)

k n +1 ∏ l=n U(t l+1 ,t l ) 1 ≤ a 2 e -t-s 2 A 1 . ( 3
For the both case the rate of convergence ε α,β (n) for approximants (3.8),(3.9) is the same as in (1.11).

Note that extension of Theorem 1.4 to Gibbs semigroups generated by a family of non-negative self-adjoint operators {A(t)} t∈I can be done along the arguments outlined in Section 2 of [START_REF] Vuillermot | A general TrotterKato formula for a class of evolution operators[END_REF]. To this end one needs to add more conditions to (A1)-(A4) that allow to control the family {A(t)} t∈I .

Here we also comment that a general scheme of the lifting due to Lemma 2.4 and Theorem 1.4 can be applied to any symmetrically-normed ideal C φ (H) of compact operators C ∞ (H), [START_REF] Zagrebnov | Gibbs Semigroups, Operator Theory Series: Advances and Applications[END_REF] Ch.6. We return to this point elsewhere.

(1, 2 )

 2 α,β > 0. 3. Since lim n→∞ k n /n = 1/2 and t > s, by definition (1.6) and by Lemma 2.4 for contractions

n- 1

 1 (t, s) × • • • × W

  )and by(2.3) we ascertain that {U(t, s)} (t,s)∈∆ ∈ C 1 (H) for t > s.Therefore, we can construct solution operator {U(t, s)} (t,s)∈∆ as a trace-norm convergent Dyson-Phillips series ∑ ∞ n=0 S n (t, s) by iteration of the integral formula (2.3) for t > s. To this aim we define the recurrence relation

  Lemma 2.1. While for t > s these arguments prove a generalisation of Theorem 2.2 and Corollary 2.3 to the trace-norm topology in Banach space C 1 (H): Theorem 2.2 Let assumptions (A1)-(A4) be satisfied. Then evolution family {U(t, s)} (t,s)∈∆ (2.9) gives for t > s a mild trace-norm continuous solution of nACP (2.1) in Banach space C 1 (H). Proof. Since by Remark 1.2(c),(d) the function t → U(t, s) for t ≥ s is strongly continuous and since U(t, s) ∈ C 1 (H) for t > s, the product U(t + δ ,t)U(t, s) is continuous in the trace-norm topology for |δ | < ts . Moreover, since {u(t)}

	Corollary 2.3 For t > s the evolution family {U(t, s)} (t,s)∈∆ (2.9) is a strict solution
	of the nACP :
	∂ t U(t, s) = -C(t)U(t, s), t ∈ (s, T ) and U(s, s) = 1, C(t) := A + B(t),	(s, T ) ⊂ [0, T ], (2.10)
	in Banach space C 1 (H).

s≤t≤T is a classical solution of nACP (1.1), equation (2.1) implies that U(t, s) has strong derivative for any t > s. Then again by Remark 1.2(d) the trace-norm continuity of δ → U(t + δ ,t)U(t, s) and by inclusion of ranges: ran(U(t, s)) ⊆ dom(A) for t > s, the trace-norm derivative ∂ t U(t, s) at t(> s) exists and belongs to C 1 (H).

Therefore, U(t, s) ∈ C((s, T ], C 1

  By virtue of Lemma 2.4 the proof of Theorem 1.4 can be carried over almost verbatim for approximants { U n (t, s)} (t,s)∈∆ ,n≥1 : (t, s) :=e -t-s n B(t k ) e -t-s n A , k = 1, 2, . . . , n, U n (t, s) := W as well as for self-adjoint approximants { U n (t, s)} (t,s)∈∆ ,n≥1 :

					.6)
	4. Since for t > s the trace-norm c(t -s) := e -t-s 2 A	1 < ∞, by (3.2)-(3.6) we
	obtain the proof of the estimate (1.11) for
			R α,β (t, s) := (a 1 R	(1) α,β + a 2 R	(2) α,β ) c(t -s) ,	(3.7)
	and 0 ≤ s < t ≤ T .
	Corollary 3.1 W	(n)	(n) n (t, s) W n-1 (t, s) × • • • × W (n) 2 (t, s) W (n) 1 (t, s), (n)	(3.8)
	W	(n)

k k (t, s) :=e -t-s n A/2 e -t-s n B(t k ) e -t-s n A/2 , k = 1, 2, . . . , n, U n (t, s) := W

Acknowledgments

This paper was motivated by my lecture at the Conference "Operator Theory and Krein Spaces" (Technische Universitäte Wien, 19-22 December 2019), dedicated to the memory of Hagen Neidhardt.

I am thankful to organisers: Jussi Behrndt, Aleksey Kostenko, Raphael Pruckner, Harald Woracek, for invitation and hospitality.