Trace formula for contractions and it's representation in \mathbb{D}
aa r X i v : . [ m a t h . F A ] F e b TRACE FORMULA FOR CONTRACTIONS AND IT’SREPRESENTATION IN D ARUP CHATTOPADHYAY AND KALYAN B. SINHA
Abstract.
The aim of this article is twofold: give a short proof of the existence of real spec-tral shift function and the associated trace formula for a pair of contractions, the differenceof which is trace-class and one of the two a strict contraction, so that the set of assumptionsis minimal in comparison to those in all the existing proofs. The second one is to find atrace formula for differences of functions of contraction and its adjoint, in which case, theintegral in the formula is over the unit disc and has an expression surprisingly similar to theHelton-Howe formula. Introduction
The notion of spectral shift function (SSF) for a trace-class perturbation of a self-adjointoperator originated in the work of Lifshitz [8], followed by Krein in [6], in which it was shownthat given a pair of (not necessarily bounded) self-adjoint operators H , H such that H − H is trace-class, there exists a unique real-valued function L ( R )-function (SSF) ξ satisfying“ Krein’s Trace Formula ”:(1.1) Tr (cid:8) φ ( H ) − φ ( H ) (cid:9) = Z R φ ′ ( λ ) ξ ( λ ) dλ, for a large class of functions φ . Krein’s original proof uses complex function theory whereasan alternative proof of (1.1) by Voiculescu [15] uses the idea of quasi-diagonalization forbounded self-adjoint operators. Later, Sinha and Mohapatra (in [10, 14]) adapted the quasi-diagonalization method to the cases of unbounded self-adjoint and unitary operators. For apair ( U, U ) of unitary operators with U − U is trace-class, Krein’s trace formula is(1.2) Tr (cid:8) φ ( U ) − φ ( U ) (cid:9) = Z π φ ′ ( θ ) ˜ ξ ( θ ) dθ = Z T φ ′ ( t ) ξ ( t ) dt, for φ : T −→ C with an absolutely convergent Fourier series for φ ′ and with the spectralshift function ξ a real-valued L ( T )-function, unique upto an additive constant. The classesof function φ given here, for which formula (1.1) and (1.2) hold, is not optimal and improvedresults in this direction was obtained by Peller and co-authors in [2].For a pair of contractions T , T with T − T is trace-class, Neidhardt [12] initiated thestudy of trace formula, to be followed by others in [1, 9]. In all these attempts, there wereadditional hypotheses involving the associated defect operators, for example all the defect Mathematics Subject Classification.
Key words and phrases.
Krein’s trace formula, Spectral shift function, Self adjoint operators, Unitaryoperators, Contractions, Unitary dilations. operators D T := ( I − T ∗ T ) , D T := ( I − T T ∗ ) , D T ∗ := ( I − T T ∗ ) , D T ∗ := ( I − T T ∗ ) were assumed to be trace-class as well in [12]. On the other hand, as an example the authors in[9] constructs a pair ( T, T ) of contractions such that T − T is trace-class, and T is Fredholm(that is, dim Ker ( T ) = dim Ker ( T ∗ ) < ∞ ) of index 0, for which a real spectral shift functionis obtained.The main purpose here is to prove the trace formula like that in (1.2) with a real spectralshift function for a pair of contractions T , T , one of which is a strict contraction (that is,either k T k or k T k <
1) and T − T is trace-class with no further assumptions as in earlierstudies . As it turns out, the trace formula involves an integral, as on the right-hand side of(1.2), which is supported on T . On the other hand, the spectra of both T and T are compactsubsets of the closed unit disc D . Clearly, it will be desirable to have a trace formula, wherethe integral will be supported on the unit disc D instead of T . For this, we need to considerfunctions of not just T and T , but functions of ( T, T ∗ ) and of ( T , T ∗ ), and we show that thetrace of the difference of such functions has an integral expression, in which the integral issupported on D (with planar Lebesgue measure), containing the natural harmonic extensionof the real spectral shift function ξ ∈ L ( T ) to ˜ ξ on D by the Poisson integral formula.Furthermore, it turns out that this integral formula bears an intriguing resemblance to theHelton-Howe formula (see [5, 3]).Here we denote by B ( H ) the collection of all bounded linear operators on H and N ( B )denotes the null space of the operator B ∈ B ( H ), B ( H ) stands for the Banach space oftrace-class operators on H and R and N for the set of real numbers and natural numbersrespectively. The rest of the paper is organized as follows: In section 2, we give the existenceof real spectral shift function for a large class of pairs of contractions ( T, T ) with trace classdifference, and section 3 deals with extending the trace formula for functions of ( T, T ∗ ) and( T , T ∗ ) with the integral over the unit disc D .2. Trace formula for contractions
In this section, a short and direct proof of the trace formula is given for a pair (
T, T ) ofcontractions, with a real-valued SSF, when T − T ∈ B ( H ) and one of the two, say T , is astrict contraction, that is 0 ≤ T < I or equivalently D T ≥ δI > δ >
0. In fact,the proof involved is symmetric with respect to the interchange of T and T . The methodemployed here is similar, to that of [9] in the sense that the Nagy-dilation of a contractionin H to a unitary in l Z ( H ) is used. However, unlike in Theorem 9.4 of [9], we do not assumethat the differences of the dilated unitaries is trace-class, rather we prove it . Lemma 2.1.
Let A and B be two positive contractions on a separable infinite-dimensionalHilbert space H . ( i ) If furthermore, N ( B ) = { } , then for every f ∈ H (2.1) (cid:0) A − B (cid:1) f = Z + ∞ e − tA (cid:0) A − B (cid:1) e − tB f dt, where the right hand side of (2.1) exists as an improper strong Riemann-Bochner integral. RACE FORMULA FOR CONTRACTIONS AND IT’S REPRESENTATION IN D ( ii ) If, B ≥ δI > for some positive δ and if (cid:0) A − B (cid:1) ∈ B ( H ) , then (cid:0) A − B (cid:1) ∈ B ( H ) .Proof. ( i ) By the definition of strong Riemann-Bochner integral [16] and by the observationthat strong- ddt n e − tA (cid:0) A − B (cid:1) e − tB f o = − e − tA (cid:0) A − B (cid:1) e − tB f, one gets that for every f ∈ H ,(2.2) (cid:0) A − B (cid:1) f − e − sA (cid:0) A − B (cid:1) e − sB f = Z s e − tA (cid:0) A − B (cid:1) e − tB f dt and for s > s ′ > − Z ss ′ e − tA (cid:0) A − B (cid:1) e − tB f dt = e − sA (cid:0) A − B (cid:1) e − sB f − e − s ′ A (cid:0) A − B (cid:1) e − s ′ B f. By hypothesis N ( B ) = { } , this implies by the spectral theorem that (cid:13)(cid:13) e − sB f (cid:13)(cid:13) = Z − e − sλ kE B ( dλ ) f k −→ s −→ + ∞ , by the dominated convergence theorem, where E B ( · ) is the spectral measure corresponding tothe operator B , and hence, the right hand side of (2.3) converges to 0 as s ′ −→ + ∞ for every f ∈ H . This proves the existence of the improper strong Riemann-Bochner integral whichappears in (2.1) as well as the equality in (2.1).( ii ) By ( i ), we have an identity in B ( H ):(2.4) A − B = Z + ∞ e − tA (cid:0) A − B (cid:1) e − tB dt, since the hypothesis in ( ii ) in particular implies that N ( B ) = { } . Moreover, the right handside integral in (2.4) converges in B ( H )-topology since (cid:13)(cid:13) e − tA (cid:13)(cid:13) ≤ (cid:13)(cid:13) e − tB (cid:13)(cid:13) ≤ e − δt . (cid:3) The following corollary is a simple consequence of the above lemma.
Corollary 2.2.
Let T and T be two contractions on a separable infinite dimensional Hilbertspace H such that one of T and T is a strict contraction and T − T ∈ B ( H ) . Then both D T − D T and D T ∗ − D T ∗ are trace class operators.Proof. Without loss of generality we assume that k T k <
1. Then it follows that there exists apositive δ ( >
0) such that D T ≥ δI > D T ∗ ≥ δI >
0. Now, by hypothesis we concludethat D T − D T = ( T − T ) ∗ T + T ∗ ( T − T ) and D T ∗ − D T ∗ = ( T − T ) T ∗ + T ( T − T ) ∗ are trace class operators and therefore the proof follows by applying Theorem 2.1 ( ii ) with A = D T and B = D T ; and A = D T ∗ and B = D T ∗ respectively. (cid:3) Now we are in a position to state and prove our main result in this section. For this, weset G := { f ( z ) = ∞ X k =0 a k z k : z ∈ D a k ∈ C and ∞ X k =0 | ka k | < ∞} . CHATTOPADHYAY AND SINHA
Theorem 2.3.
Let T and T be two contractions on a separable infinite dimensional Hilbertspace H such that T is a strict contraction and T − T ∈ B ( H ) . Then there exists a unique(up to an additive constant) real valued function ξ ∈ L ([0 , π ]) such that (2.5) Tr H (cid:8) φ ( T ) − φ ( T ) (cid:9) = Z π ddt (cid:8) φ (e it ) (cid:9) ξ ( t ) dt for every φ in G .Proof. Using the construction of the Sch¨ a ffer matrix unitary dilation of contractions (see [11],chapter I, section 5), we dilate T and T to the corresponding unitary operators U T and U T respectively on the same Hilbert space ℓ Z ( H ), that is T n = P H U nT | H and T n = P H U nT | H for n ≥ , where P H is the orthogonal projection of ℓ Z ( H ) onto H and furthermore the explicit expres-sions of U T and U T are as follows: U T (cid:0) { x n } n ∈ Z (cid:1) := n · · · , x − , x − , D T x − T ∗ x , T x + D T ∗ x , x , x , . . . o , { x n } n ∈ Z ∈ ℓ Z ( H ) , (2.6)and U T (cid:0) { x n } n ∈ Z (cid:1) := n · · · , x − , x − , D T x − T ∗ x , T x + D T ∗ x , x , x , . . . o , { x n } n ∈ Z ∈ ℓ Z ( H ) , (2.7)where the boxed entry in the above sequences corresponds to the term indexed by 0 and weidentify the Hilbert space H as a closed subspace of ℓ Z ( H ) consisting of sequences { x n } n ∈ Z such that x n = 0 for n = 0. Thus for { x n } n ∈ Z ∈ ℓ Z ( H ) we have (cid:0) U T − U T (cid:1)(cid:0) { x n } n ∈ Z (cid:1) := n · · · , , , ( D T − D T ) x − ( T ∗ − T ∗ ) x , ( T − T ) x + (cid:0) D T ∗ − D T ∗ (cid:1) x , , , · · · o . (2.8)In other words, there will be exactly four non-zero entries namely T − T at the (0 ,
0) position, D T ∗ − D T ∗ at the (0 ,
1) position, D T − D T at the ( − ,
0) position and − ( T ∗ − T ∗ ) at the( − ,
1) position survive in the block matrix representation of U T − U T . On the other hand byhypothesis since T is a strict contraction and T − T is trace class, then from the Corollary 2.2we conclude that both D T − D T and D T ∗ − D T ∗ are trace class operators. Therefore itfollows from (2.8) that all non-zero entries in the block matrix representation of U T − U T are trace class and hence U T − U T is also a trace class operator in ℓ Z ( H ). This implies that U nT − U nT ∈ B ( ℓ Z ( H )) for all n ≥
1. Moreover, it is also clear from equations (2.6) and (2.7)that both U T and U T are upper triangular matrices with the only non-zero diagonal entries T and T and hence both U nT and U nT are also upper triangular matrices with the only non-zerodiagonal entries T n and T n respectively for any n ≥
1. Thus T n − T n = P H { U nT − U nT }| H , and Tr H (cid:8) T n − T n (cid:9) = Tr ℓ Z ( H ) (cid:8) U nT − U nT (cid:9) , n ≥ , RACE FORMULA FOR CONTRACTIONS AND IT’S REPRESENTATION IN D and hence p ( T ) − p ( T ) = P H { p ( U T ) − p ( U T ) }| H , andTr H (cid:8) p ( T ) − p ( T ) (cid:9) = Tr ℓ Z ( H ) (cid:8) p ( U T ) − p ( U T ) (cid:9) (2.9)for any polynomial p ( · ) in D . Next for φ ( z ) = ∞ P k =0 a k z k ∈ G , and if we denote p n ( z ) = n P k =0 a k z k ,then it is easy to check that p n ( T ), p n ( T ) and p n ( U T ), p n ( U T ) converge to φ ( T ), φ ( T ) and φ ( U T ), φ ( U T ) respectively in operator norm. Therefore from the above equation (2.9) weconclude that(2.10) φ ( T ) − φ ( T ) = P H { φ ( U T ) − φ ( U T ) }| H . Furthermore, since ∞ P k =0 | ka k | < ∞ as φ ∈ G and U T − U T ∈ B ( ℓ Z ( H )) , it follows that (cid:13)(cid:13)(cid:13)(cid:8) φ ( U T ) − φ ( U T ) (cid:9) − (cid:8) p n ( U T ) − p n ( U T ) (cid:9)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k = n +1 a k (cid:8) U kT − U kT (cid:9)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k = n +1 a k n k − X j =0 U k − j − T ( U T − U T ) U jT o(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k U T − U T k ∞ X k = n +1 | ka k | −→ n −→ ∞ . (2.11)Thus (cid:8) φ ( U T ) − φ ( U T ) (cid:9) ∈ B ( ℓ Z ( H )) and from (2.10) we conclude that (cid:8) φ ( T ) − φ ( T ) (cid:9) is atrace class operator and(2.12) Tr H (cid:8) φ ( T ) − φ ( T ) (cid:9) = Tr ℓ Z ( H ) n P H { φ ( U T ) − φ ( U T ) } P H o . Furthermore, since (cid:8) φ ( U T ) − φ ( U T ) (cid:9) ∈ B ( ℓ Z ( H )), and since all the diagonal entries of P ⊥H (cid:8) φ ( U T ) − φ ( U T ) (cid:9) P ⊥H in the Sch¨ a ffer-dilation basis are zero, it follows thatTr ℓ Z ( H ) n P ⊥H { φ ( U T ) − φ ( U T ) } P ⊥H o = 0and hence we get(2.13) Tr H (cid:8) φ ( T ) − φ ( T ) (cid:9) = Tr ℓ Z ( H ) (cid:8) φ ( U T ) − φ ( U T ) (cid:9) . Note that U T and U T are two unitary operators on l Z ( H ) such that U T − U T ∈ B ( ℓ Z ( H )) andtherefore by Krein’s trace formula corresponding to the pair ( U T , U T ) [2, 7, 9, 10] there existsa real-valued L ([0 , π ])-function ξ (known as the SSF, corresponding to the pair ( U T , U T ))such that(2.14) Tr ℓ Z ( H ) (cid:8) φ ( U T ) − φ ( U T ) (cid:9) = Z π ddt (cid:8) φ (e it ) (cid:9) ξ ( t ) dt for every φ in G . Finally, combining (2.13) and (2.14) we arrive at:Tr H (cid:8) φ ( T ) − φ ( T ) (cid:9) = Z π ddt (cid:8) φ (e it ) (cid:9) ξ ( t ) dt (2.15) CHATTOPADHYAY AND SINHA for every φ in G . Since ξ ∈ L ( T ), real-valued, it follows that if there is another such η suchthat (2.15) is satisfied with η replacing ξ , then this will imply that Z π ddt (cid:8) φ (e it ) (cid:9) (cid:2) ξ ( t ) − η ( t ) (cid:3) dt = 0 for every φ ∈ G , in particular Z π in e int (cid:2) ξ ( t ) − η ( t ) (cid:3) dt = 0 ∀ n ≥ . Taking the complex-conjugate of this relation, we arrive at Z π e int (cid:2) ξ ( t ) − η ( t ) (cid:3) dt = 0 ∀ n ∈ Z \ { } . Since ξ − η ∈ L ( T ), this would imply that ξ − η = constant or that ξ is unique modulo anadditive constant. This completes the proof. (cid:3) Remark 2.4. ( i ) The SSF ξ for a pair ( U, U ) of unitaries is real-valued L ( T ) -function, byits construction ( [6, 9, 10] ). However, while one can write Tr (cid:8) U n − U n (cid:9) = Z π in e int ξ ( t ) dt for all n ∈ Z one can not, in general, do the same for (cid:8) T n − T n (cid:9) for n a negative integer because T (or T ) need not be invertible. On the other hand, the same purpose will be served if one takesthe adjoint instead and note that: if k T k ≤ , k T k < and T − T ∈ B ( H ) , the same is truefor the pair ( T ∗ , T ∗ ) replacing the pair ( T, T ) . This leads to (recalling that ξ is real-valued) Tr (cid:8) T ∗ n − T ∗ n (cid:9) = Tr (cid:8) T n − T n (cid:9) = Z π in e int ξ ( t ) dt = − in Z π e − int ξ ( t ) dt, (2.16) for n ∈ N , though the functions (cid:8) e − int | n ∈ N (cid:9) do not belong to G . We shall exploit thissimple observation in the next section to construct a different kind of trace formula for thedifference of functions of ( T, T ∗ ) and of ( T , T ∗ ) to obtain an integral expression with supporton D instead of on T . ( ii ) As we have observed earlier that, for n ∈ N T ∗ n − T ∗ n = P H { U ∗ n T − U ∗ n T }| H = P H { U − nT − U − nT }| H . But instead one could have dilated the pair ( T ∗ , T ∗ ) of contractions to obtain for n ∈ N , T ∗ n = P H U nT ∗ | H , T ∗ n = P H U nT ∗ | H . Thus though Tr H (cid:8) T ∗ n − T ∗ n (cid:9) = Tr ℓ Z ( H ) (cid:8) U ∗ T n − U ∗ n T (cid:9) = − in Z π e − int ξ ( t ) dt for n ∈ N , it is also equal to Tr ℓ Z ( H ) (cid:8) U nT ∗ − U nT ∗ (cid:9) = in Z π e int χ ( t ) dt, RACE FORMULA FOR CONTRACTIONS AND IT’S REPRESENTATION IN D with both ξ and χ are real-valued L ( T ) -functions. It may be noted that while (cid:8) U T ∗ − U T ∗ (cid:9) is an upper triangular matrix-operator, (cid:0) U ∗ T − U ∗ T (cid:1) = (cid:0) U T − U T (cid:1) ∗ is a lower-triangular one.Thus, we do not expect the two SSF’s ξ and χ to be related. However, it is easy to see that: − in Z π e − int ξ ( t ) dt = in Z π e int χ ( t ) dt or ˆ χ ( n ) = − ˆ ξ ( − n ) for each n ∈ N \ { } or equivalently χ ( t ) + ξ ( − t ) = constant . Trace formula with support on the disc D The SSF ξ in (1.1) corresponding to a pair of self-adjoint operators ( H, H ) such that H − H ∈ B ( H ) is supported on a subset of R which contains the spectrum of H as well asthe spectrum of H . The same is true for SSF η in (1.2) corresponding to a pair of unitaryoperators ( U, U ) with trace class difference, that is, the support of η lies in T and containthe spectrum of U and U . Therefore it is expected while dealing with a pair of contractions( T, T ) with T − T ∈ B ( H ), the shift function corresponding to that pair ( T, T ) should alsobe supported on a subset of the closed unit disc D which contains the spectrum of T and T .But in (2.5) we see that this is not the case, that is, the shift function ξ corresponding to thepair ( T, T ) is supported on the unit circle T whereas the spectrum of T and T is containedin D . Our next part of the analysis is devoted to obtaining an appropriate justification of thefact mentioned above. Next observe that the Lebesgue measure in C , restricted to T is zeroand therefore, to obtain a shift function supported on D and satisfying a formula like (2.5)corresponding to a pair of contractions ( T, T ), it is necessary to consider functions of theoperator along with its adjoint instead of that of the operator alone. To obtain an extensionof L ( T )-function as a harmonic function into the interior D of T , we next use the the Poissonintegral representation. Let f ∈ L ( T ), then the Poisson integral of f is denoted by P f andis defined by(3.1) (
P f )( z ) = 12 π Z π − | z | | e it − z | f (e it ) dt, where z ∈ D and dt π is the normalized Lebesgue measure on T . Now one can verify that for z ∈ D ,(3.2) 1 − | z | | e it − z | = 1 + ∞ X n =1 ¯ z n e int + ∞ X n =1 z n e − int , and therefore combining equations (3.1) and (3.2) we conclude that P f is a harmonic functionon D and that(3.3) ( P f )( z ) = ˆ f (0) + ∞ X n =1 ˆ f ( − n )¯ z n + ∞ X n =1 ˆ f ( n ) z n , for z ∈ D , where ˆ f ( n ) is the n -the Fourier coefficient of f given byˆ f ( n ) = 12 π Z π f (e it ) e − int dt, where n ∈ Z . CHATTOPADHYAY AND SINHA
In other words, through Poisson integral transform, we extend an L ( T )- function defined onthe boundary to the interior of T , that is on D as a harmonic function. One of the oldestresults about the boundary behavior of P f is due to Fatou, and it says that the originalfunction f is retrieved almost everywhere as a boundary value of P f . Theorem 3.1.
Let f ∈ L ( T ) , then lim r −→ − ( P f )( r e it ) = f (e it ) for all e it ∈ T except possibly on a set of measure zero. For more on the Poisson integral and related matter see [4, 13] and the references therein.Note that we prove our main theorem (see Theorem 3.3) in this section for the following classof functions e G := n ψ : T −→ C (cid:12)(cid:12)(cid:12) ∞ X n = −∞ | n ˆ ψ ( n ) | < ∞ , where ˆ ψ ( n ) = 12 π Z π ψ (e it ) e − int dt, n ∈ Z o . Let ψ ∈ e G and set for z ∈ D = D ∪ T ,(3.4) e ψ ( z, ¯ z ) = ˆ ψ (0) + ∞ X n =1 ˆ ψ ( − n )¯ z n + ∞ X n =1 ˆ ψ ( n ) z n . Now it is important to observe that the Poisson integral transform of ψ , that is P ψ maynot exists for z ∈ T but the right hand side of (3.4) makes sense for z ∈ T because of theextra assumption on ψ , namely ∞ P n = −∞ | n ˆ ψ ( n ) | < ∞ . This implies that ∞ P n = −∞ | ˆ ψ ( n ) | < ∞ andfurthermore the equation (3.3) implies that the extension (3.4) of ψ to the interior D of T issame as the Poisson integral transform of ψ , that is e ψ ( z, ¯ z ) = ( P ψ )( z ) for z ∈ D . Next weneed the following useful lemma towards obtaining our main result in this section. Lemma 3.2.
Let f : T −→ C be such that ∞ P n = −∞ | ˆ f ( n ) | < ∞ , and let g ∈ L ([0 , π ]) . Then π Z π f (e it ) g ( t ) dt = ∞ X n = −∞ ˆ f ( n ) ˆ g ( − n ) . Proof.
Since ∞ P n = −∞ | ˆ f ( n ) | < ∞ , then it is easy to verify that f is a bounded continuous functionon T and furthermore we have that(3.5) f (e it ) = ∞ X n = −∞ ˆ f ( n ) e int , RACE FORMULA FOR CONTRACTIONS AND IT’S REPRESENTATION IN D where the series in the right hand side converges uniformly on T . Thus using the aboveexpression (3.5) of f we conclude that12 π Z π f (e it ) g ( t ) dt = 12 π Z π ∞ X n = −∞ ˆ f ( n ) e int ! g ( t ) dt = ∞ X n = −∞ ˆ f ( n ) (cid:18) π Z π g ( t ) e int dt (cid:19) = ∞ X n = −∞ ˆ f ( n ) ˆ g ( − n ) , where we have applied Fubini’s theorem to get the second equality since ∞ P n = −∞ | ˆ f ( n ) | < ∞ and g ∈ L ([0 , π ]). This completes the proof. (cid:3) Suppose T ∈ B ( H ) be such that k T k ≤ ψ ∈ e G weset e ψ ( T, T ∗ ) = ˆ ψ (0) I + ∞ X n =1 ˆ ψ ( − n ) T ∗ n + ∞ X n =1 ˆ ψ ( n ) T n , where the right hand side converges in operator norm since ∞ P n = −∞ | ˆ ψ ( n ) | < ∞ . Now we are ina position to state and prove our main result in this section for a class of function e G . Theorem 3.3.
Let T and T be two contractions on a separable infinite dimensional Hilbertspace H such that T is a strict contraction and T − T ∈ B ( H ) . Then for ψ ∈ e G , the operator e ψ ( T, T ∗ ) − e ψ ( T , T ∗ ) is trace class and (3.6) Tr H n e ψ ( T, T ∗ ) − e ψ ( T , T ∗ ) o = Z D J (cid:0)e ξ, e ψ (cid:1) ( z, ¯ z ) dz ∧ d ¯ z, where e ψ is as in (3.4) , ξ is the shift function corresponding to the pair ( T, T ) as in (2.5) and e ξ ( z, ¯ z ) = ( P ξ )( z ) for z ∈ D , J (cid:0)e ξ, e ψ (cid:1) = ∂ e ξ∂z ∂ e ψ∂ ¯ z − ∂ e ψ∂z ∂ e ξ∂ ¯ z is the Jacobian of e ξ and e ψ on D , dz ∧ d ¯ z is the Lebesgue measure on D and the integral in the right hand side of (3.6) is to beinterpreted as an improper Riemann-Lebesgue integral lim R ↑ Z { z || z |≤ R< } J (cid:0)e ξ, e ψ (cid:1) ( z, ¯ z ) dz ∧ d ¯ z. Proof.
First we note that e ψ ( T, T ∗ ) − e ψ ( T , T ∗ ) = ∞ X n =1 ˆ ψ ( − n ) (cid:8) T ∗ n − T ∗ n (cid:9) + ∞ X n =1 ˆ ψ ( n ) (cid:8) T n − T n (cid:9) = ∞ X n =1 n − X j =0 ˆ ψ ( − n ) T ∗ n − j − ( T ∗ − T ∗ ) T ∗ j + ∞ X n =1 n − X j =0 ˆ ψ ( n ) T n − j − ( T − T ) T j . (3.7) Since T − T ∈ B ( H ) and ∞ P n = −∞ | n ˆ ψ ( n ) | < ∞ because ψ ∈ ˜ G , then both the series in the righthand side of the above equation (3.7) converge in trace norm and hence e ψ ( T, T ∗ ) − e ψ ( T , T ∗ )is trace class and furthermore we have (cid:13)(cid:13) e ψ ( T, T ∗ ) − e ψ ( T , T ∗ ) (cid:13)(cid:13) ≤ ∞ X n = −∞ | n ˆ ψ ( n ) | ! k T − T k < ∞ , and(3.8) Tr H n e ψ ( T, T ∗ ) − e ψ ( T , T ∗ ) o = ∞ X n =1 ˆ ψ ( − n ) Tr H n T ∗ n − T ∗ n o + ∞ X n =1 ˆ ψ ( n ) Tr H n T n − T n o . Next by combining (2.16) and (3.8) we getTr H n e ψ ( T, T ∗ ) − e ψ ( T , T ∗ ) o = ∞ X n =1 ˆ ψ ( − n ) − in Z π e − int ξ ( t ) dt + ∞ X n =1 ˆ ψ ( n ) in Z π e int ξ ( t ) dt = 2 πi ∞ X n =1 ˆ ψ ( − n ) ( − n ) ˆ ξ ( n ) + ∞ X n =1 ˆ ψ ( n ) ( n ) ˆ ξ ( − n )= 2 πi ∞ X n = −∞ n ˆ ψ ( n ) ˆ ξ ( − n ) = 2 π ∞ X n = −∞ ˆ ψ ′ ( n ) ˆ ξ ( − n )which by applying Lemma 3.2 corresponding to f = ψ ′ and g = ξ yields(3.9) Tr H n e ψ ( T, T ∗ ) − e ψ ( T , T ∗ ) o = Z π ddt (cid:8) ψ (e it ) (cid:9) ξ ( t ) dt = 2 πi ∞ X n = −∞ n ˆ ψ ( n ) ˆ ξ ( − n ) . Also since for z ∈ D , | z | <
1, the Jacobian has the expression, J (cid:0)e ξ, e ψ (cid:1) = ∂ e ξ∂z ∂ e ψ∂ ¯ z − ∂ e ψ∂z ∂ e ξ∂ ¯ z = ∞ X n =1 n ˆ ξ ( n ) z n − ! ∞ X m =1 m ˆ ψ ( − m )¯ z m − ! − ∞ X m =1 m ˆ ξ ( − m )¯ z m − ! ∞ X n =1 n ˆ ψ ( n ) z n − ! = ∞ X n,m =1 nm (cid:8) ˆ ξ ( n ) ˆ ψ ( − m ) − ˆ ξ ( − m ) ˆ ψ ( n ) (cid:9) z n − ¯ z m − , RACE FORMULA FOR CONTRACTIONS AND IT’S REPRESENTATION IN D where the double series converges absolutely and uniformly for | z | ≤ R <
1. Therefore for
R < Z D R J (cid:0)e ξ, e ψ (cid:1) dz ∧ d ¯ z = Z D R ∞ X n,m =1 nm (cid:8) ˆ ξ ( n ) ˆ ψ ( − m ) − ˆ ξ ( − m ) ˆ ψ ( n ) (cid:9)! z n − ¯ z m − dz ∧ d ¯ z = ∞ X n,m =1 nm (cid:8) ˆ ξ ( n ) ˆ ψ ( − m ) − ˆ ξ ( − m ) ˆ ψ ( n ) (cid:9) Z D R z n − ¯ z m − dz ∧ d ¯ z, (3.10)where we have used Fubini’s theorem to interchange the summation and integration because ∞ X n,m =1 (cid:12)(cid:12) nm (cid:8) ˆ ξ ( n ) ˆ ψ ( − m ) − ˆ ξ ( − m ) ˆ ψ ( n ) (cid:9)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z D R z n − ¯ z m − dz ∧ d ¯ z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π ∞ X n,m =1 nm (cid:8) | ˆ ξ ( n ) || ˆ ψ ( − m ) | + | ˆ ξ ( − m ) || ˆ ψ ( n ) | (cid:9) Z R r n + m − r dr ≤ π k ξ k L ([0 , π ]) ∞ X n = −∞ | n ˆ ψ ( n ) | ! ∞ X n,m =1 nmn + m R n + m < ∞ . Finally from (3.10) we get Z D R J (cid:0)e ξ, e ψ (cid:1) dz ∧ d ¯ z = − i ∞ X n,m =1 nm (cid:8) ˆ ξ ( n ) ˆ ψ ( − m ) − ˆ ξ ( − m ) ˆ ψ ( n ) (cid:9) × Z R Z π r n + m − e i ( n − m ) t rdrdt = − πi ∞ X n,m =1 nm (cid:8) ˆ ξ ( n ) ˆ ψ ( − m ) − ˆ ξ ( − m ) ˆ ψ ( n ) (cid:9) (cid:18) R n + m n + m (cid:19) δ nm = 2 πi ∞ X n = −∞ n ˆ ψ ( n ) ˆ ξ ( − n ) R n and hence lim R ↑ Z D R J (cid:0)e ξ, e ψ (cid:1) dz ∧ d ¯ z = 2 πi ∞ X n = −∞ n ˆ ψ ( n ) ˆ ξ ( − n )(3.11)since ∞ P n = −∞ | n ˆ ψ ( n ) | < ∞ and | ˆ ξ ( n ) | ≤ k ξ k L ([0 , π ]) < ∞ for all n ∈ Z . Thus the conclusion ofthe theorem follows by combining equations (3.9) and (3.11). This completes the proof. (cid:3) Acknowledgements
The first author (AC) acknowledges the support from the Mathematical Research ImpactCentric Support (SERB) project by the Department of Science & Technology (DST), G.O.I,and the second author (KBS) thanks Indian National Science Academy for its support throughthe Senior Scientist Scheme.
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Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039,India
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