THE EXPONENTIAL OF QUASI BLOCK-TOEPLITZ MATRICES*

Elahe BOLOURCHIAN Bijan Ahmadi KAKAVANDI

Department of Applied Mathematics，F(xiàn)aculty of Mathematical Sciences，Shahid Beheshti University，Tehran，Iran

E-mail:elahebolourchian@gmail.com;b ahmadi@sbu.ac.ir

Abstract The matrix Wiener algebra，WN:=MN(W) of order N＞0，is the matrix algebra formed by N×N matrices whose entries belong to the classical Wiener algebra W of functions with absolutely convergent Fourier series.A block-Toeplitz matrix T (a)=[Ai，j]i，j≥0 is a block semi-in finite matrix such that its blocks Ai，j are finite matrices of order N，Ai，j=Ar，s whenever i-j=r-s and its entries are the coefficients of the Fourier expansion of the generator a:T→MN(C).Such a matrix can be regarded as a bounded linear operator acting on the direct sum of N copies of L2(T).We show that exp (T (a)) differes from T (exp (a)) only in a compact operator with a known bound on its norm.In fact，we prove a slightly more general result:for every entire function f and for every compact operator E，there exists a compact operator F such that f (T (a)+E)=T (f (a))+F.We call these T (a)+E′ s matrices，the quasi block-Toeplitz matrices，and we show that via a computation-friendly norm，they form a Banach algebra.Our results generalize and are motivated by some recent results of Dario Andrea Bini，Stefano Massei and Beatrice Meini.

Key words Toeplitz matrix;in finite matrix;block matrix;exponential;functional calculus

1 Introduction

A Toeplitz matrix[ai，j]is a matrix in which，its entries are constant along diagonals;that is，ai，j=ar，swheneveri-j=r-s.Leta∈L∞(T) be a bounded Lebesgue measurable function，where T={t∈C:|t|=1}={eiθ:θ∈R}is the unit circle in the complex plane，and consider its Fourier coefficients

One can define the semi-in finite matrixA=[ai，j]i，j∈Z+by lettingai，j:=ai-jfor alli，j∈Z+={0，1，2，...}，called the Toeplitz operator associated witha.Also，we use notationT(a):=Ato emphasizing this dependence.It is known thatAacts on the Banach space?2:=?2(Z+) as a bounded linear operator.The functionais called the generator or the symbol function ofA.Now，for any positive integerN，the truncated matrixAN:=[ai，j]0≤i，j＜Nis a Toeplitz matrix of orderN×N.

After Otto Toeplitz (1881–1940)，the study of the so-called Toeplitz operators and Toeplitz matrices has grown rapidly and became a vast area of investigation by many authors，due to their rich theoretical aspects in Functional Analysis and Linear Algebra as well as their profound and ubiquitous applications in many concrete problems of engineering.There are many good books and monographs that the interested readers could be referred to，for example，[11，20]and also[10，12].

Let W be the Wiener algebra which is the algebra of all functionsa:T→C of absolutely convergent Fourier series.It is known that for eacha∈W，there exists some compact operatorFsuch that exp (T(a))=T(exp (a))+F;see[8].The perturbation of a Toeplitz operator by compact ones，that is，‘T(a)+a compact operator’is called a quasi-Toeplitz operator.Furthermore，the exponential of any quasi-Toeplitz operator is a quasi-Toeplitz operator again.Dario Andrea Bini et al.suggested some numerical algorithms for computing the exponential of quasi-Toeplitz operator;see[6–8].Also，there is a generalization of this fact to some functions other than the exponential function[5].

In this paper，motivating by the mentioned works，we try to generalize these results to block-Toeplitz and quasi block-Toeplitz matrices.For this aim，we used the notion of matrix Wiener algebra;see Section 2.1 below.By a finite (respectively semi-in finite) block matrix A=[Ai，j]，we mean a finite (respectively semi-in finite) matrix such that all of its blocksAi，jare finite，sayN×Nmatrices.Block-Toeplitz matrices are block matrices such thatAi，j=Ar，swheneveri-j=r-s.This kind of matrices appear in signal processing and some other fields;for example，see[19]，[30，Chapter 8]and[24].For mathematical aspects of these class of matrices，one can refer to[12，Chapter 6]and[25].Also，we offer a somehow stronger version of a known result about the exponential of quasi-Toeplitz matrices;see Remark 6.3.It is worthwhile to mention that there is a discrete asymptotical counterpart of the theory and results of this paper in the theory of Generalized Locally Toeplitz (GLT) matrix sequences.We refer the reader to[17，18]for the basic GLT theory in the scalar case，unilevel and multilevel and to[2，3]for the GLT block-case，unilevel and multilevel.Also，[16]is a nice and interesting paper on matrix functions and their applications within this framework.

The paper is organized as follows:definitions of matrix Wiener algebra WN，block-Toeplitz，block-Hankel and quasi block-Toeplitz matrices are presented in Section 2.Proposition 2.2 and Theorem 2.4 are the main assertions of this section，which altogether assert that the product of two block-Toeplitz matrices is a quasi block-Toeplitz matrix.In Section 3，we introduce two Banach algebras of quasi block-Toeplitz matrices.Also，using a so-called‘higher order’of matrix Wiener algebra W′N，a computation-friendly norm is introduced (see Proposition 3.6).Sections 4 and 5 are devoted to the exponential of block-Toeplitz and quasi block-Toeplitz matrices，respectively.The results of these two sections are generalized to some more general functions，for example，entire functions in Section 6.Finally in Section 7，a technical Proposition 7.1 is presented which makes it possible to approximate the block semi-in finite matrices by finite matrices.

2 Basic Definitions and Concepts

2.1 Wiener matrix algebra

Let T={t∈C:|t|=1}={eiθ:θ∈R}be the complex unit circle.Recall that the classical Wiener algebra W is the Banach algebra of all functionsf:T→C with absolutely convergent Fourier coefficients.More precisely，W consists of all functions

Notice that since ‖f‖∞≤‖f‖Wfor allf∈W，we have W?L∞(T).

Now，for any positive integerN，the Wiener matrix algebra WN:=MN(W) is the Banach algebra of allN×Nmatrices with entries in W.Hence，an element of WNcan be interpreted as a matrix functiona:T→MN(C) whose entries belong to W;for example，see[11，15]for more details.A noteworthy feature is that one may consider the matrix Fourier expansion ofa∈W

where ‖am‖2is the usual matrix norm

for eachA∈MN(C).Using the inequality ‖AB‖2≤‖A‖2‖B‖2for any twoN×NmatricesAandB，and the Cauchy product of two power series，we get

for alla，b∈WN;that is，the norm ‖·‖WNis sub-multiplicative as we desired.Notice that ‖a(t)‖2≤‖a‖WNfor alla∈WNand allt∈T.Hence，in the sense of Bttcher and Silbermann[12]，we have

2.2 Block-Toeplitz matrix

First，let us introduce some useful notations.For each，let

wheret∈T.Notice thata+，a-，and～abelong to WNwhenevera∈WNand～a=a(1t) for allt∈T.Now，as we will see，one can associate to eacha∈WNa semi-in finite block matrix with a block-Toeplitz structure.

Definition 2.1Everya∈WNgenerates a semi-in finite block-Toeplitz matrix briefly called BT matrix，defined by

where{ak}k∈Zis the sequence of matrix Fourier coefficients ofa;see (2.1) and (2.2).

The matrix functionais called the symbol ofT(a).In the caseN=1，T(～a) is the transposed operator ofT(a).Beside this，the block-Hankel matrix，brie fly called BH matrix，generated byaand～a，are given by

respectively.

2.3 Semi-in finite block matrix as operator

First，recall that any semi-in finite matrix can be considered as a linear operator on an appropriate linear space.More precisely，a semi-in finite matrixA=[ak，l]k，l∈Z+acts on the classical Banach space?2by definition:Ax=yif and only if

wherex={xj}j∈Z+，y={yj}j∈Z+∈?2.The domain of this operator，that is，thosex’s in?2such thaty∈?2too，may be only a subspace of?2.However，it is possible to consider the in finite matrices (2.9) and (2.10) as operators on?2，but it will be more convenient to study block matrices withN×Nblocks on the CN-valued?2-space，that is，?2Nis the space of all vector-valued sequencesx:N→CNsuch that

for anya∈WN?MN(L∞(T))，and so they are bounded operators;see[12，Chapter 6].Here，the notation ‖·‖opdenotes the operator norm，that is，

2.4 Quasi block-Toeplitz matrix

Quasi block-Toeplitz matrices are compact perturbations of block-Toeplitz matrices.Recall that a compact operatorKon a Hilbert space H is a bounded linear operator such that it is a norm-limit of a sequence of finite-rank operators.Also，we know that the set of all compact operators is a norm-closed two-sided ideal in the algebra of all bounded linear operators B (H);for example，see[31].

Proposition 2.2Every BH semi-in finite matrix is a compact operator acting on the Banach space?2N.

The proof can be found in[12，p.188].

Definition 2.3Let KNbe the set of all semi-in finite block matrices withN×Nblocks such that they are compact operators acting on?2Nas linear operators.Then we call any operatorA=T(a)+Ea semi-in finite quasi block-Toeplitz matrix，briefly called a QBT matrix，wherea∈WNandE∈KN.

The following key fact can be deduced from Proposition 2.2 and the well-known identity (6.2) on page 188 of[12]:

Theorem 2.4The product of two BT semi-in finite matrices is a QBT semi-in finite matrix.In fact，T(ab)=T(a)T(b)+H(a)H(～b) for eacha，b∈MN(L∞(T)).

3 Banach Algebras of QBT Matrices

Theorem 2.4 implies that the space of all QBT operators is a Banach sub-algebra of B ();but，this norm is not appropriate for computational purposes.In order to find a computationfriendly norm on the space of all QBT operators which makes it a Banach algebra again，and imitating the D.A.Bini et al.’s idea，we introduce a new norm as follows:

Letα＞0，a∈WNandE∈KN.Define

We show that this norm is a sub-multiplicative norm provided thatTo verify this，letA=T(a)+EaandB=T(b)+Ebbe two QBT operators，wherea，b∈WN，and letC:=ABandc:=ab.By Theorem 2.4，we have thatC=T(c)+Ec，where

is compact since the sum of products of operators of which at least a factor is compact.So，

as we desired.We denote this algebra by A.On the other hand，the mappingψ:A→WN×KNby definitionψ(T(a)+E)=(αa，E) is a norm-preserving bijection between A and the Banach algebra WN×KN，so A is a Banach algebra too.One can summarize the above discussion as follows:

Proposition 3.1The space A equipped with following norm is a Banach algebra:

Here is an alternative algebraic norm.First notice that formally for each

Lemma 3.2The spaceis a Banach algebra.

ProofFirst，we have to verify the sub-multiplication property of the norm.Leta，b∈.We have

Here we used (2.4).The completeness ofis deduced from (2.5)，the completeness of WN，and a classical theorem in Mathematical Analysis which asserts that if ‖an-a‖∞→0 andasn→∞，thenb=a′;see，for example，[32，Theorem 7.17]. □

Also，consider the following norm on block matrixE=(ei，j)i，j∈Z+:

Also，it deserves to mention that this norm is sub-multiplicative and

for any matrixA∈MN(C);see，for example，[23].

Lemma 3.3The space F:={E|Eis a block matrix with ‖E‖F(xiàn)＜+∞}equipped with the ‖·‖F(xiàn)-norm ‖·‖F(xiàn)is a Banach algebra.

ProofAgain，this is another straightforward verification.For example，let us writeE=[ei，j]i，j∈Z+，F(xiàn)=[fi，j]i，j∈Z+andEF=[hi，j]i，j∈Z+，whereHence

The completeness follows from the completeness of the classical Banach space?1(S) for every arbitrary setS. □

Lemma 3.4Leta，b∈andc:=ab.ThenT(a)T(b)=T(c)+Efor some compact operatorE∈F∩KN.Moreover

ProofBy Theorem 2.4，we haveT(a)T(b)=T(c)+E，whereE=H(a)H(～b) is a compact operator.Notice thatH(a)=H(a+) andH(～b)=H(b-).Using (3.1)，we have

Substitutingawithbcompletes the proof. □

Lemma 3.5Leta∈WNandE∈F∩KN.Then we have

ProofBy sub-multiplicative property of the ‖·‖F(xiàn)-norm，we have

Now，consider the space of all QBT semi-in finite matrices with generators in，that is，

Proposition 3.6The space B equipped with following norm is a Banach algebra

ProofFirst，notice that the mappingby definition

is a bijection，which preserves the norms，so B is a Banach space since the product space×F is.On the other hand，assume thatA=T(a)+EaandB=T(b)+Ebbelong to B and letc:=ab.Then，we have

by Theorem 2.4，where

Sincec∈by Lemma 3.2，we have to show that ‖Ec‖F(xiàn)＜+∞.By Lemmas 3.3，3.4 and 3.5，we know that ‖EaEb‖F(xiàn)，‖T(a)Eb‖F(xiàn)and ‖H(a+)H(b-)‖F(xiàn)are finite.On the other hand，sincefor allF∈F，we have

Therefore by (3.3)，‖Ec‖F(xiàn)＜+∞.For sub-multiplicativity property，we write

4 The Exponential of BT Matrices

Recall that for any bounded operator，finite or semi-in finite matrixA，one can define its exponential by

for every sub-multiplicative norm ‖·‖.

In the case BT semi-in finite matrixT(a)，wherea∈WN，we get

by (4.1)，(2.12) and sub-multiplicative property of operator norm.

It is known that the exponential of any quasi-Toeplitz matrix is a quasi-Toeplitz matrix again;see[6，7]and[8].In this paper，we show that the exponential of any QBT matrix is a QBT matrix too.More precisely，we prove that for anya∈WNand each compact operatorE∈KN，there exists a compact operatorF∈KNsuch that exp (T(a)+E)=T(exp (a))+F;see Theorems 5.1 and 5.2 below.In this section，we treat the case of BT matrices，that is，the caseE=0.

Proposition 4.1Leta∈WNand defineEi:=T(a)i-T(ai)，fori≥1.Then

Also，for everyi≥2，the operatorsEiare compact and

ProofBy definition ofEiand Theorem 2.4，we have

fori≥2，as we desired.For the norm estimation，fixi≥2，and using the sub-multiplicative property，we have

Since the compact operators form a closed ideal in B ()，the compactness ofEi’s follows by induction. □

Now，we can prove that the exponential of any BT matrix is a QBT matrix.

Theorem 4.2Givena∈WN，we have

ProofFor any integerk≥1，let

The operatorFis compact because it is a limit of compact operators.Finally，

Here，as an alternative，we focus on B，that is，operatorsT(a)+E，wherea∈andE∈F∩KN.First，we express a technical lemma.

Lemma 4.3Leta∈.Then we have

fork≥1.

ProofAs we mentioned before，we have

On the other hand，

fork≥1. □

Now，the following statement can be stated:

Theorem 4.4For anya∈W′Nandi≥2，we have

Moreover，exp (T(a))=T(exp (a))+F，whereFis given by Theorem 4.2 and

ProofBy Lemmas 3.5，4.3 and Proposition 4.1，we have

for alli≥2.Notice thatE0=E1=0.Using the fact that...and by induction，we have

which completes the proof. □

5 Exponential of QBT Matrices

In this section，we generalize the results in previous section to QBT semi-in finite matrices.

Theorem 5.1LetA=T(a)+Ebe a QBT matrix，wherea∈WNand letEbe a compact operator acting onThen，exp (A)=T(exp (a))+F，whereFis a compact operator onand

ProofLetF=exp (A)-T(exp (a)).Obviously，

In order to prove the compactness ofF，let

Fori≥0，we have thatDi:=(T(a)i-T(ai))+G，whereGis a compact operator，(since each term of it has at least a factor ofE)，and using Proposition 4.1，the compactness of eachDifollows.Sincethe conclusion follows. □

Now，in addition，ifA=T(a)+E∈B，that is，a∈andE∈F∩KN，then one can improve the statement of Theorem 5.1.For this aim，using Theorem 2.4 and (5.1)，we have

fori≥1.Hence，we obtain

Notice that every term of this equation is compact，and recall that by definition，we have ‖K‖B=‖K‖F(xiàn)for every compact operatorK.Now，taking B-norm and using Lemmas 3.5 and 4.3，we obtain

Theorem 5.2IfA=T(a)+E∈B，then exp (A)=T(exp (a))+F，whereFis a compact operator and

6 Functional Calculus

In this section，we want to generalize the results of previous sections to some appropriate functions beyond the exponential map.In fact，we consider holomorphic functions of BT and QBT matrices.

Theorem 6.1Letbe the Laurent series of an entire function，that is，a functionf:C→C which is holomorphic at all finite points over the whole complex plane C and define the real-valued functionF:[0，+∞)→[0，+∞) byFor everya∈WN，there exists a compact operatorR∈KNsuch that

ProofThe functionf0(z):=f(z)-f(0)(z∈C) is also an entire function and

for allb∈WN，that is，Tis a bounded linear operator from WNto B ().Hence，

for all convergent sequence{xk}?WNin ‖·‖WN-norm.Now，using the identity

and the monotonicity property mentioned above，we get

Finally，we have

in operator norm.Now，compactness ofRis deduced from the compactness of the operatorsEi’s in Proposition 4.1 and the norm-closedness of the subspace of all compact operators in B (). □

Remark 6.3It is interesting that not only Theorem 6.1 is a generalization of Theorem 4.2，but also improves it.In fact，estimation (6.1) in Example 6.2 is much better than the estimation (4.3) in Theorem 4.2.Of course，this is a better result even in the semi-in finite Toeplitz matrices setting.

Now，we survey the case ofA=T(a)+E∈QBT.We provide conditions under whichf(A) is well defined and is a QBT matrix too.

Theorem 6.4LetA=T(a)+E∈A be a QBT matrix and letbe an analytical function on{z∈C;|z|＜ρ}for someρ＞0.If ‖A‖A＜ρa(bǔ)nd ‖a‖WN＜ρ，thenis a QBT matrix too，and we havef(A)=T(f(a))+Ffor some compact operatorF.

ProofThe proof is completely similar to the proof of Theorem 4.2.First，notice that since

we havef(a)∈WN.Now，letAk=T(ak)+Ek，whereEk:=T(a)k-T(ak) fork≥1.Then we haveS(k)(A)=G(k)+F(k)，where as beforeandFinally，we getand

which is a compact operator because it is a limit of compact operators. □

Moreover，we have a similar result for the algebra B (see Proposition 3.6).

Theorem 6.5LetA=T(a)+E∈B，that is，a∈，E∈F∩KNand letbe an analytical function on{z∈C;|z|＜ρ}for someρ＞0.If ‖A‖B＜ρ，thenis well-defined andf(A)=T(f(a))+G，whereG∈F∩KN.

ProofRecall that any power series absolutely converges inside the convergence disk;hence the seriesconverges as ‖A‖B＜ρby assumption.Now，using the inequalitythe convergence of the seriesciAiin the Banach algebra B follows from the Cauchy criteria.

Sincef(A)∈B，it can be written asf(A)=T(b)+Gfor some functionb∈and some compact operatorG∈F.Letbe the sequence of the partial sums off(z).Then，there is a sequence of compact operators{Ek}?F such thatfk(A)=T(fk(a))+Ek.Thus，

Using definition of the norm ‖·‖B，we conclude that

Hence，b=f(a)，as we desired. □

7 Approximation

Recall that in Section 3，we introduced some computation-friendly norms，in which make the space of all QBT matrices to be a Banach algebra;see Propositions 3.1 and 3.6.In fact，we proved that the spaces A={T(a)+E|a∈WN，E∈KN}，and B={T(a)+E|a∈W′N，E∈F∩KN}，equipped with following norms:

respectively，are two Banach algebras，where KNwas defined in Definition 2.3.In order to numerically approximate QBT semi-in finite matrices，we have to approximate them by finite matrices.LetA=[ai，j]i，j∈Z+be a semi-in finite block matrix with blocksOne can cut this semi-in finite block matrix to reduce it to a finite block matrix as follows:Given positive integerm，let

Using these notations，we have the following statement:

Proposition 7.1It is given that∈＞0，a functiona∈WNand a compact operatorE∈KN.Then for sufficiently largem，there exists a semi-in finite block matrixFsuch that forA=T(a)+EandB=T(a)(m)+F(m)，we have

ProofRecall (2.1)，(2.2) and (2.3).Sincethere exists positive integermsuch that

Notice that (7.2) remains true for all integers larger thanm.

On the other hand，sinceEis a compact operator，it can be approximated by finite rank matrices.For this end，first choose two sequences of unit vectors{uk}，{vk}inand choose a sequence of complex numbers{σk}such thatin the norm topology.So，for sufficiently largen≥m，we have

Here，we have used the following fact:

Therfore，using (7.3) and (7.4)，we get

Now，without loss of generality，we may assumem=m′.Also，notice that since all of the entries vanish aftermNfirst components，the matixcan be interpreted as anmN×mNmatrix.Finally，lettingB=T(a)(m)+F(m)and using (7.2) and (7.5)，we have

Proposition 7.2Given∈＞0，a functiona∈and a compact operatorE∈KN，then，for sufficiently largem，there exists a semi-in finite block matrixGsuch that forA=T(a)+EandC=T(a)(m)+G(m)，we have

ProofThis is similar to the proof of Proposition 7.1 and so it is omitted.Just note that

8 Conclusion

Using the notion of the matrix Wiener algebra，we generalized most of recent results of D.A.Bini et al.regarding exponential of semi-in finite (quasi) Toeplitz matrices[4–7]and[8]to the exponential of semi-in finite (quasi) block-Toeplitz matrices.Furthermore，besides the exponential operator，we also consider some analytic or entire functions to reformulate new results.Finally，we suggest an approximation method for numerical purposes;but more effective numerical methods should be investigated in future works.