Solution Operator for Non-Autonomous Perturbation of Gibbs Semigroup
aa r X i v : . [ m a t h . F A ] J a n Solution Operator for Non-AutonomousPerturbation of Gibbs Semigroup
To the memory of Hagen Neidhardt
Valentin A. ZagrebnovAix-Marseille Universit´e, CNRS, Centrale Marseille, I2MInstitut de Math´ematiques de Marseille (UMR 7373)CMI - Technopˆole Chˆateau-Gombert39 rue F. Joliot Curie, 13453 Marseille, France
Abstract
The paper is devoted to a linear dynamics for non-autonomous perturbation of theGibbs semigroup on a separable Hilbert space. It is shown that evolution fam-ily { U ( t , s ) } ≤ s ≤ t solving the non-autonomous Cauchy problem can be approxi-mated in the trace-norm topology by product formulae. The rate of convergenceof product formulae approximants { U n ( t , s ) } { ≤ s ≤ t , n ≥ } to the solution operator { U ( t , s ) } { ≤ s ≤ t } is also established. The aim of the paper is two-fold. Firstly, we study a linear dynamics, which isa non-autonomous perturbation of Gibbs semigroup. Secondly, we prove productformulae approximations of the corresponding to this dynamics solution operator { U ( t , s ) } { ≤ s ≤ t } , known also as evolution family , fundamental solution , or propaga-tor , see [1] Ch.VI, Sec.9.To this end we consider on separable Hilbert space H a linear non-autonomousdynamics given by evolution equation of the type: ∂ u ( t ) ∂ t = − C ( t ) u ( t ) , u ( s ) = u s , s ∈ [ , T ) ⊂ R + , C ( t ) : = A + B ( t ) , u s ∈ H , t ∈ I : = [ , T ] , (1.1)where R + = { } ∪ R + and linear operator A is generator of a Gibbs semigroup. Notethat for the autonomous Cauchy problem (ACP) (1.1), when B ( t ) = B , the outlined programme corresponds to the Trotter product formula approximation of the Gibbssemigroup generated by a closure of operator A + B , [20] Ch.5.The main result of the present paper concerns the non-autonomous Cauchy prob-lem (nACP) (1.1) under the following Assumptions: (A1) The operator A ≥ in a separable Hilbert space H is self-adjoint. The fam-ily { B ( t ) } t ∈ I of non-negative self-adjoint operators in H is such that the boundedoperator-valued function ( + B ( · )) − : I −→ L ( H ) is strongly measurable.(A2) There exists α ∈ [ , ) such that inclusion: dom ( A α ) ⊆ dom ( B ( t )) , holdsfor a.e. t ∈ I . Moreover, the function B ( · ) A − α : I −→ L ( H ) is strongly measurableand essentially bounded in the operator norm: C α : = ess sup t ∈ I k B ( t ) A − α k < ∞ . (1.2)(A3) The map A − α B ( · ) A − α : I −→ L ( H ) is H¨older continuous in the operatornorm: for some β ∈ ( , ] there is a constant L α , β > k A − α ( B ( t ) − B ( s )) A − α k ≤ 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 ≥ , thatis, a strongly continuous semigroup such that G ( t ) | t > ∈ C ( H ) . Here C ( H ) denotesthe ∗ -ideal of trace-class operators in C ∗ -algebra L ( H ) of bounded operators on H . Remark 1.1
Assumptions (A1)-(A3) are introduced in [4] to prove the operator-norm convergence of product formula approximants { U n ( t , s ) } ≤ s ≤ t to solution op-erator { U ( t , s ) } ≤ s ≤ t . Then they were widely used for product formula approxima-tions in [10]-[15] in the context of the evolution semigroup approach to the nACP,see [6]-[9] . Remark 1.2
The following main facts were established ( see, e.g., [4, 7, 18, 19]) 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 semi-groups. Hence, the nACP (1.1) is of parabolic type [5, 16] . (b) Since domains dom ( C ( t )) = dom ( A ) , t ≥ , 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, uniformlybounded family of operators { U ( t , s ) } ( t , s ) ∈ ∆ , ∆ : = { ( t , s ) ∈ I × I : 0 ≤ s ≤ t ≤ T } ,such that the conditionsU ( t , t ) = 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) . on-Autonomous Dynamics 3 (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 approxi-mants { U n ( t , s ) } ( t , s ) ∈ ∆ , n ≥ to solution operator { U ( t , s ) } ( t , s ) ∈ ∆ . Let s = t < t < . . . < t n − < t n < t , t k : = s + ( k − ) t − sn , (1.5)for k ∈ { , , . . ., n } , n ∈ N , be partition of the interval [ s , t ] . Then correspondingapproximants may be defined as follows: W ( n ) k ( t , s ) : = e − t − sn A e − t − sn B ( t k ) , k = , , . . . , n , U n ( t , s ) : = W ( n ) n ( t , s ) W ( n ) n − ( t , s ) × · · · × W ( n ) ( t , s ) W ( n ) ( t , s ) . (1.6)It turns out that if the assumptions (A1)-(A3), adapted to a Banach space X , aresatisfied for α ∈ ( , ) , β ∈ ( , ) and in addition the condition α < β holds, thensolution operator { U ( t , s ) } ( t , s ) ∈ ∆ admits the operator-norm approximationess sup ( t , s ) ∈ ∆ k U n ( t , s ) − U ( t , s ) k ≤ 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 ≥ is determined by the smoothness of the perturbation B ( · ) in (A3)and by the parameter of inclusion in (A2), see [13].The Lipschitz case β = X in [10]. There it was shown thatif α ∈ ( / , ) , then one gets estimateess sup t ∈ I k U n ( t , s ) − U ( t , s ) k ≤ R , α n − α , n = , , . . . . (1.8)For the Lipschitz case in a Hilbert space H the assumptions (A1)-(A3) yield astronger result [4]:ess sup ( t , s ) ∈ ∆ k U n ( t , s ) − U ( t , s ) k ≤ R log ( n ) n , n = , , . . . . (1.9)Note that actually it is the best of known estimates for operator-norm rates of con-vergence under conditions (A1)-(A3).The estimate (1.7) was improved in [12] for α ∈ ( / , ) in a Hilbert space usingthe evolution semigroup approach [2, 3, 9]. This approach is quite different fromtechnique used for (1.9) in [4], but it is the same as that employed in [10]. Proposition 1.3 [12]
Let assumptions (A1)-(A3) be satisfied for β ∈ ( , ) . If β > α − > , then estimate ess sup ( t , s ) ∈ ∆ k U n ( t , s ) − U ( t , s ) k ≤ R β n β , (1.10) V. A. Zagrebnov holds for n ∈ N and for some constant R β > . Note that the condition β > α − β > α (1.7), but it does not coverthe Lipschitz case (1.8) because of condition β < any known operator-normbounds (1.7)-(1.10) (we denote them by R α , β ε α , β ( n ) ) to estimate in the trace-normtopology k · k . This is a subtle matter even for ACP, see [20] Ch.5.4.- The first step is the construction for nACP (1.1) a trace-norm continuous solutionoperator { 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 k U n ( t , s ) − U ( t , s ) k . Theorem 1.4
Let assumptions (A1)-(A4) be satisfied. Then the estimate k U n ( t , s ) − U ( t , s ) k ≤ R α , β ( t , s ) ε α , β ( n ) , (1.11) holds for n ∈ N and ≤ s < t ≤ T for some R α , β ( t , s ) > . Besides Remark 1.2(a)-(d) we also recall the following assertion, see, e.g., [16],Theorem 1, [17], Theorem 5.2.1.
Proposition 2.1
Let assumptions (A1)-(A3) be satisfied. (a)
Then solution operator { U ( t , s ) } ( t , s ) ∈ ∆ is strongly continuously differentiable for ≤ s < t ≤ T and ∂ t U ( t , s ) = − ( A + B ( t )) U ( t , s ) . (2.1)(b) Moreover, the unique function t u ( t ) = U ( t , s ) u s is a classical solution of (1.1) for initial data H s = dom ( A ) . Note that solution of (1.1) is called classical if u ( t ) ∈ C ([ , T ] , H ) ∩ C ([ , T ] , H ) , u ( t ) ∈ dom ( C ( t )) , u ( s ) = u s , and C ( t ) u ( t ) ∈ C ([ , T ] , H ) for all t ≥ s , with conven-tion that ( ∂ t u )( s ) is the right-derivative, see, e.g., [16], Theorem 1, or [1], 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 k U ( t , s ) u k = − ( C ( t ) U ( t , s ) u , U ( t , s ) u ) ≤ , 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 ) − Z ts d τ G ( t − τ ) B ( τ ) U ( τ , s ) , U ( s , s ) = . (2.3) on-Autonomous Dynamics 5 Hence evolution family { U ( t , s ) } ( t , s ) ∈ ∆ , which is defined by equation (2.3), can beconsidered as a mild solution of nACP (2.1) for 0 ≤ s ≤ t ≤ T in the Banach space L ( H ) of bounded operators, cf. [1], Ch.VI.7.Note that assumptions (A1)-(A4) yield for 0 ≤ s < t ≤ T , τ ∈ ( s , t ) and for theclosure A − α B ( τ ) : k G ( t − s ) A α k ≤ M α ( t − τ ) α and k A − α B ( τ ) k ≤ C α . (2.4)Then (2.2), (2.4) give the trace-norm estimate (cid:13)(cid:13)(cid:13)(cid:13) Z ts d τ G ( t − τ ) B ( τ ) U ( τ , s ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ M α C α − α ( t − s ) − α , (2.5)and by (2.3) we ascertain that { U ( t , s ) } ( t , s ) ∈ ∆ ∈ C ( H ) for t > s .Therefore, we can construct solution operator { U ( t , s ) } ( t , s ) ∈ ∆ as a trace-norm convergent Dyson-Phillips series ∑ ∞ n = S n ( t , s ) by iteration of the integral formula(2.3) for t > s . To this aim we define the recurrence relation S ( t , s ) = U A ( t − s ) , S n ( t , s ) = − Z ts d s G ( t − τ ) B ( τ ) S n − ( τ , s ) , n ≥ . (2.6)Since in (2.6) the operators S n ≥ ( t , s ) are the n -fold trace-norm convergent Bochnerintegrals S n ( t , s ) = Z ts d τ Z τ s d τ . . . Z τ n − s d τ n G ( t − τ )( − B ( τ )) G ( τ − τ ) · · · G ( τ n − − 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 ) − α / ( − α ) = : ξ < k S n ( t , s ) k ≤ ξ n , n ≥ . (2.8)Consequently ∑ ∞ n = S n ( t , s ) converges for t > s in the trace-norm and satisfies theintegral equation (2.3). Thus we get for solution operator of nACP the representation U ( t , s ) = ∞ ∑ n = 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 thenext Theorem 2.2 and Corollary 2.3, but only in the strong ([19] Proposition 3.1,main Theorem in [18]) and in the operator-norm topology, [4] Lemma 2.1. Whilefor t > s these arguments prove a generalisation of Theorem 2.2 and Corollary 2.3to the trace-norm topology in Banach space C ( H ) : V. A. Zagrebnov
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 ( H ) . Corollary 2.3
For t > s the evolution family { U ( t , s ) } ( t , s ) ∈ ∆ (2.9) is a strict solutionof the nACP : ∂ t U ( t , s ) = − C ( t ) U ( t , s ) , t ∈ ( s , T ) and U ( s , s ) = , C ( t ) : = A + B ( t ) , ( s , T ) ⊂ [ , T ] , (2.10) in Banach space C ( H ) .Proof. Since by Remark 1.2(c),(d) the function t U ( t , s ) for t ≥ s is stronglycontinuous and since U ( t , s ) ∈ C ( H ) for t > s , the product U ( t + δ , t ) U ( t , s ) iscontinuous in the trace-norm topology for | δ | < t − s . Moreover, since { u ( t ) } s ≤ t ≤ T is a classical solution of nACP (1.1), equation (2.1) implies that U ( t , s ) has strongderivative 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 ( H ) .Therefore, U ( t , s ) ∈ C (( s , T ] , C ( H )) ∩ C (( s , T ] , C ( H )) with U ( s , s ) = and U ( t , s ) ∈ C ( H ) , C ( t ) U ( t , s ) ∈ C ( H ) for t > s , which means that solution U ( t , s ) of (2.10) is strict , cf. [19] Definition 1.1. (cid:3) We note that these results for ACP in Banach space C ( H ) are well-known forGibbs semigroups, see [20], Chapter 4.Now, to proceed with the proof of Theorem 1.4 about trace-norm convergence ofthe 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 ( H ) for t > ,and let V , V , . . . , V n be bounded operators L ( H ) . Then (cid:13)(cid:13)(cid:13) n ∏ j = V j e − t j A (cid:13)(cid:13)(cid:13) ≤ n ∏ j = k V j kk e − ( t + t + ... + t n ) A / k , (2.11) for any set { t , t , . . . , t n } of positive numbers.Proof. At first we prove this assertion for compact operators: V j ∈ C ∞ ( H ) , j = , , . . . , n . Let t m : = min { t j } nj = > T : = ∑ nj = t j >
0. For any 1 ≤ j ≤ n , wedefine an integer ℓ j ∈ N by 2 ℓ j t m ≤ t j ≤ ℓ j + t m . Then we get ∑ nj = ℓ j t m > T / n ∏ j = V j e − t j A = n ∏ j = V j e − ( t j − ℓ j t m ) A ( e − t m A ) ℓ j . (2.12) on-Autonomous Dynamics 7 By the definition of the k ·k -norm and by inequalities for singular values { s k ( · ) } k ≥ of compact operators (cid:13)(cid:13)(cid:13) n ∏ j = V j e − t j A (cid:13)(cid:13)(cid:13) = ∞ ∑ k = s k (cid:0) n ∏ j = V j e − ( t j − ℓ j t m ) A ( e − t m A ) ℓ j (cid:1) ≤ ∞ ∑ k = n ∏ j = s k (cid:16) e − ( t j − ℓ j t m ) A (cid:17) (cid:2) s k ( e − t m a ) (cid:3) ℓ j s k ( V j ) ≤ ∞ ∑ k = s k ( e − t m A ) ∑ nj = ℓ j n ∏ j = k V j k . (2.13)Here we used that s k ( e − ( t j − ℓ j t m ) A ) ≤ k e − ( t j − ℓ j t m ) A k ≤ s k ( V j ) ≤ k V j k . Let N : = ∑ nj = ℓ j and T m : = Nt m > T /
2. Since ∞ ∑ k = s k ( e − tA / q ) q = ( k e − tA / q k q ) q , the inequality (2.13) yields for q = N : (cid:13)(cid:13)(cid:13) n ∏ j = V j e − t j A (cid:13)(cid:13)(cid:13) ≤ (cid:16)(cid:13)(cid:13) e − T m A / N (cid:13)(cid:13) N (cid:17) N n ∏ j = k V j k . (2.14)Now we consider an integer p ∈ N such that 2 p ≤ N < p + . It then follows that T / < T m / < p T m / N , and hence we obtain (cid:16)(cid:13)(cid:13) e − T m A / N (cid:13)(cid:13) q = N (cid:17) N = ∞ ∑ k = s Nk ( e − T m A / N ) (2.15) ≤ ∞ ∑ k = s p k ( e − p T m A / p N ) ≤ ∞ ∑ k = s p k ( e − TA / p + ) = k e − TA / k , where we used that s k ( e − T m A / N ) = s k ( e − p T m A / p N ) ≤ k e − T m A / N k ≤
1, and that s k ( e − ( t + τ ) A ) ≤ k e − tA k s k ( e − τ A ) ≤ s k ( e − τ A ) for any t , τ >
0. Therefore, the estimates(2.14), (2.15) give the bound (2.11).Now, let V j ∈ L ( H ) , j = , , . . . , n , and set ˜ V j : = V j e − ε A for 0 < ε < t m . Hence,˜ V j ∈ C ( H ) ⊂ C ∞ ( H ) and s k ( ˜ V j ) ≤ k ˜ V j k ≤ k V j k . If we set ˜ t j : = t j − ε , then (cid:13)(cid:13)(cid:13) n ∏ j = V j e − t j A (cid:13)(cid:13)(cid:13) ≤ n ∏ j = k V j kk e − ( ˜ t + ˜ t + ··· + ˜ t n ) A / k . (2.16)Since the semigroup { e − tA } t ≥ is k · k -continuous for t >
0, we can take in (2.16)the limit ε ↓
0. This gives the result (2.11) in general case. (cid:3)
V. A. Zagrebnov
We follow the line of reasoning of the lifting lemma developed in [20], 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 ) = ∏ k = n W ( n ) k ( t , s ) − ∏ l = n U ( t l + , t l ) . (3.1)Let integer k n ∈ ( , n ) . Then (3.1) yields the representation: U n ( t , s ) − U ( t , s ) = k n + ∏ k = n W ( n ) k ( t , s ) − k n + ∏ l = n U ( t l + , t l ) ! ∏ k = k n W ( n ) k ( t , s )+ k n + ∏ l = n U ( t l + , t l ) ∏ k = k n W ( n ) k ( t , s ) − ∏ l = k n U ( t l + , t l ) ! , which implies the trace-norm estimate k U n ( t , s ) − U ( t , s ) k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k n + ∏ k = n W ( n ) k ( t , s ) − k n + ∏ l = n U ( t l + , t l ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∏ k = k n W ( n ) k ( t , s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k n + ∏ l = n U ( t l + , t l ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∏ k = k n W ( n ) k ( t , s ) − ∏ l = k n U ( t l + , t l ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (3.2)2. Now we assume that lim n → ∞ k n / n = /
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 ) ∈ ∆ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k n + ∏ k = n W ( n ) k ( t , s ) − U ( t , ( t + s ) / ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ R ( ) α , β ε α , β ( n ) , (3.3)ess sup ( t , s ) ∈ ∆ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∏ k = k n W ( n ) k ( t , s ) − U (( t + s ) / , s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ R ( ) α , β ε α , β ( n ) , (3.4)for n ∈ N and for some constants R ( , ) α , β > n → ∞ k n / n = / t > s , by definition (1.6) and by Lemma 2.4 forcontractions { V k = e − t − sn B ( t k ) } nk = there exists a > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∏ k = k n W ( n ) k ( t , s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∏ k = k n e − t − sn A e − t − sn B ( t k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ a k e − t − s A k . (3.5)Similarly there is a > on-Autonomous Dynamics 9 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k n + ∏ l = n U ( t l + , t l ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ a k e − t − s A k . (3.6)4. Since for t > s the trace-norm c ( t − s ) : = k e − t − s A k < ∞ , by (3.2)-(3.6) weobtain the proof of the estimate (1.11) for R α , β ( t , s ) : = ( a R ( ) α , β + a R ( ) α , β ) c ( t − s ) , (3.7)and 0 ≤ s < t ≤ T . (cid:3) Corollary 3.1
By virtue of Lemma 2.4 the proof of Theorem 1.4 can be carried overalmost verbatim for approximants { b U n ( t , s ) } ( t , s ) ∈ ∆ , n ≥ : b W ( n ) k ( t , s ) : = e − t − sn B ( t k ) e − t − sn A , k = , , . . . , n , b U n ( t , s ) : = b W ( n ) n ( t , s ) b W ( n ) n − ( t , s ) × · · · × b W ( n ) ( t , s ) b W ( n ) ( t , s ) , (3.8) as well as for self-adjoint approximants { e U n ( t , s ) } ( t , s ) ∈ ∆ , n ≥ : e W ( n ) k ( t , s ) : = e − t − sn A / e − t − sn B ( t k ) e − t − sn A / , k = , , . . . , n , e U n ( t , s ) : = e W ( n ) n ( t , s ) e W ( n ) n − ( t , s ) × · · · × e W ( n ) ( t , s ) e W ( n ) ( t , s ) . (3.9) For the both case the rate of convergence ε α , β ( n ) for approximants (3.8),(3.9) is thesame as in (1.11) . Note that extension of Theorem 1.4 to Gibbs semigroups generated by a familyof non-negative self-adjoint operators { A ( t ) } t ∈ I can be done along the argumentsoutlined in Section 2 of [18]. 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 andTheorem 1.4 can be applied to any symmetrically-normed ideal C φ ( H ) of compactoperators C ∞ ( H ) , [20] Ch.6. We return to this point elsewhere. Acknowledgments
This paper was motivated by my lecture at the Conference ”Operator Theory andKrein Spaces” (Technische Universit¨ate Wien, 19-22 December 2019), dedicated tothe memory of Hagen Neidhardt.I am thankful to organisers: Jussi Behrndt, Aleksey Kostenko, Raphael Pruckner,Harald Woracek, for invitation and hospitality.
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