Generalized convolution operators
andasymptoticspectraltheory
von der Fakultät für Mathematik der Technischen Universität Chemnitz genehmigte Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr. rer. nat.)
vorgelegt von Dipl.-Math. Olga Nikolaievna Zabroda geboren am 29. Oktober 1979 in Azov (Russland) eingereicht am 29.06.2006
Gutachter:
Prof. Dr. B. Silbermann, TU Chemnitz Prof. Dr. T. Ehrhardt, University of California, Santa Cruz Prof. Dr. S. Grudsky, CINVESTAV, Mexico
Tag der Verteidigung: 11.12.2006
Bibliographic description
Zabroda, Olga Nikolaievna Generalized convolution operators and asymptotic spectral theory
Dissertation (in English), Chemnitz University of Technology, Department of Mathematics, Chemnitz, 2006 138 pages
Abstract The present dissertation contributes to the further advancement of the theory of various classes of discrete and continuous (integral) convolution operators. This theory is already highly developed. The most important questions in this theory are about the invertibility of convolution operators, about the construction of their inverses, and about the properties of their spectra. These problems hardly allow an explicit description. In order to tackle them in practice, one usually resorts to the ﬁnite section method. Instead of considering the invertibility of the operator itself, one tries to study the asymptotic invertibility of the sequence of its truncations, which means that one has to study the stability of this approximation sequence.
If{AN}is a sequence of matrices (of growing size) which converge strongly to a certain operatorA, then the question about the asymptotic properties of the matricesANap-pear. These questions are, as a rule, very complicated. For instance the spectrum is not continuous with respect to the Hausdorﬀ metric even in the case of convergence in the norm.
One of the probably ﬁrst non-trivial classic results which describes the asymptotic pro-perties of the spectrum of an approximation sequence goes back to G. Szegö in 1915 [99]: Leta:R−→Rbe a 2π-periodic function belonging toL1(0,2π). Then for each continuous functionf:R−→Rwith a compact support the asymptotic formula Z 2π 1 1 trf(TN(a)) =f(a(x))dx+o(1), N→ ∞ N2π0
holds. ThereinTN(a) := Fourier coeﬃcients of the TN(a) are in general not that a bounded Toeplitz ∞ belonging toL(0,2π).
(ban−k)n,k=1,...,Nare ﬁnite Toeplitz matrices constituted by the functiona. Notice that in the above setting the Toeplitz matrices the truncations of a bounded Toeplitz operators. The reason is + operatorT(a) onl2(Z) has a generating function necessarily
The study of ﬁnite sections (truncations) of bounded Toeplitz operators started with a celebrated paper by G. Baxter in 1963 (see [7]), where the stability problem (equivalently, + the problem of asymptotic invertibility) of Toeplitz operators acting inl1(Z) was com-
pletely solved. These two results initiated a bulk of further investigations which cannot be described in short. The thesis itself is devoted to the study of sequences of matrices or operators which are built up in special ways from generalized discrete or continuous (integral) convolution ope-rators. The generating function depend on three variables and this leads to considerably more complicated approximation sequences.
Key words Toeplitz operator, convolution operator, spectral theory, asymptotic behavior of the spec-trum, asymptotic invertibility, Szegö limit theorem, Banach algebra.
Contents
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2
3
4
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Introduction
7
General information on the theory of ordinary and generalized convo-lution operators 10 2.1 Discrete convolution operators and Szegö limit theorems . . . . . . . . . . 10 2.2 Integral convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Generalized convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
A Szegö type theorem for a generalized discrete convolution in the ﬁnite case 17 3.1 The formulation of the Szegö type theorem . . . . . . . . . . . . . . . . . . 17 1 3.2 A local estimate of the operatorA(a18. . . . . . . . . . . . . . . . . . . ) . N 1 3.3 The functionφ(A(a), d. . . . . . . . . . . . . . . . . . . . . . . . . . . ) . 20 N 3.4 The theorem on the almost inverse operator . . . . . . . . . . . . . . . . . 22 1 3.5 The proof of the Szegö type theorem for the operatorA(a27. . . . . . . ) . N 3.6 Some remarks to variable-coeﬃcient block Toeplitz matrices in the ﬁnite case 29
A Szegö type theorem for a generalized discrete convolution in the in-ﬁnite case 31 4.1 The formulation of the Szegö type theorem . . . . . . . . . . . . . . . . . . 31 2 4.2 A local estimate for the operatorA(a. . . . . . . . . . . . . . . . . . . ) . 32 N 2 4.3 The functionφ(A(a), d32. . . . . . . . . . . . . . . . . . . . . . . . . . . ) . N 4.4 The theorem on the almost inverse operator . . . . . . . . . . . . . . . . . 33 2 4.5 The proof of the Szegö type theorem for the operatorA(a. . . . . . . ) . 38 N 4.6 Variable-coeﬃcient block Toeplitz matrices in the inﬁnite case . . . . . . . 41
A Szegö type theorem for a generalized integral convolution in the ﬁnite case 43 5.1 The formulation of the Szegö type theorem . . . . . . . . . . . . . . . . . . 43 5.2 Some information on operator ideals . . . . . . . . . . . . . . . . . . . . . 44 5.3 The Krein algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.4 The functions Φ(a, d) and Φ(A, d47. . . . . . . . . . . ) and their properties −1−1 5.5 The operatorsC(a),C(a),B(a) andB(a48. . . . . . . . . . . . . ) . +−+− 1 5.6 A local estimate of the operatorA(a. . . . . . . . . . . . . . . . . . . ) . 50 τ 5.7 The theorem on the almost inverse operator . . . . . . . . . . . . . . . . . 51 −1 e 1 1 5.8 The asymptotic behavior of the trace of the operator (A(a))−B(a) . 54 τ δ,τ 1 5.9 An asymptotic representation of the operatorf(A(a. . . . . . . . . . 63)) . τ 5.10 The proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.11 The case of a matrix-valued symbol . . . . . . . . . . . . . . . . . . . . . . 65
A Szegö type theorem for a generalized integral convolution in the inﬁ-nite case 67 6.1 The formulation of the Szegö type theorem . . . . . . . . . . . . . . . . . . 67
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6.2 6.3 6.4 6.5 6.6 6.7 6.8
2 The function Φ(A(a), d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . τ 2 A local estimate for the operatorA(a) . . . . . . . . . . . . . . . . . . . . τ The theorem on the almost inverse operator . . . . . . . . . . . . . . . . . −1 e 2 2 The asymptotic behavior of the trace of the operator (A(a))−B(a) . τ δ,τ 2 An asymptotic representation of the operatorf(A(a)) . . . . . . . . . . . τ The proof of Theorem 6.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . Some remarks on a matrix-valued symbol . . . . . . . . . . . . . . . . . . .
68 69 69 74 80 81 82
An algebraic approach to the study of generalized discrete convolutions in the ﬁnite case 84 1 1∗1 7.1 The strong convergence of the operatorsA(a), (A(a)W N(N Na)) ,WNAN 1∗ ))W andWN(AN(aN86. . . . . in the case of a smooth generating function . 1 7.2 Algebraic properties of the sequence{A(a)}in the case of a smooth gen-N erating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.3 The stability of an operator sequence. The algebrasF,F/NandF/J. . 93 1 7.4 The norm of{A(a)}+Jin the algebraF/Jin the case of a smooth N generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 1 7.5 The norm of{A(a)}+Nin the algebraF/Nin the case of a smooth N generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.6 A Szegö type theorem for a sequence of self-adjoint operators . . . . . . . . 98 7.7 The Szegö type theorem for arbitrary sequences . . . . . . . . . . . . . . . 102
An algebraic approach to the study of generalized discrete convolutions in the inﬁnite case 104 e 8.1 The algebraF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2 2∗ 8.2 The strong limits of the operatorsA(a), (A(athe sets)) and N N 2 2∗ S-Lim ({A(a)}), S-Lim ({(A(a))}) in the case of a smooth generating N N function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2 8.3 Algebraic properties of the sequence{A(a)}in the case of a smooth gen-N erating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2 2 2 a)}+ or{A(a)} ∈Sub({A(a)}) in the 8.4 The norm of{AN(J({Ni}) fNiN i algebraF({Ni})/J({Ni}) in the case of a smooth generating function . . . 113 2 2 8.5 The algebrasAandA. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 J 2 e 8.6 The norm of{A(a)}+Nin the algebraF/Nin the case of a smooth N generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.7 A Szegö type theorem for a sequence of self-adjoint operators . . . . . . . . 118 8.8 The Szegö type theorem for arbitrary sequences . . . . . . . . . . . . . . . 122
Conclusions
References
Selbständigkeitserklärung (in Deutsch/in German)
Thesen (in Deutsch/in German)
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126
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135
Lebenslauf (in Deutsch/in German)
Danksagung (in Deutsch/in German)
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1
Introduction
The description of spectral properties and in particular the question about the invertibility of linear operators are important problems in operator theory. Their investigation is especially diﬃcult in the case of operators acting on inﬁnite-dimensional spaces. When trying to gain some answers to these questions it is in practice often convenient to study the ﬁnite section method for these operators. This method consists in considering the sequence of ﬁnite truncations of the operator instead of the operator itself. If, for instance, we want to study the spectral properties of a discrete convolution operator (Toeplitz operator), then the ﬁnite trunctations are the ﬁnite Toeplitz matrices TN(a) deﬁned by (abn−k)n,k=1,...,N. Hereabn(n∈Z) are the Fourier coeﬃcients of a functiona(t) deﬁned on the unit circle. This function is called the symbol of the sequence{TN(a)}N∈Nof the ﬁnite sections. The study of the spectral properties of the ﬁnite sections{TN(a)}N∈NasNapproaches inﬁnity goes back as early as to the 1915 paper of Szegö [99] in which he established the so-called ﬁrst Szegö Limit Theorem. One version of this theorem says that for a suﬃciently smooth functionfthe following asymptotic formula holds: NZ X 1 1 1 trf(TN(a)) =f(λj) =f(a(t))dµ+o(1) asN→ ∞(1) N N2πT j=1 Thereinλ1, ...,λNare all eigenvalues of the operatorTN(a) taking multiplicities into account, andµis the Lebesgue measure onT1952 Szegö [100] generalized this. In asymptotic formula to the so-called strong Szegö Limit Theorem and computed the second term in the asymptotics. The results obtained by Szegö marked the beginning of a series of investigations which were concentrated on such discrete convolution operators and their continuous (or integral) analogues (Wiener-Hopf operators). Chapter 2 contains some general information on the development of this theory. This dissertation presents results which describe the asymptotics of the spectrum of generalized discrete and integral convolution operators. Generalized convolution operators have appeared so far only in few papers, the most of which have been published recently. We refer in this connection to [28], [29], [49], [55], [76], [79], [89], [98], [110], [92], [93], [94], [111], [95], [62], [96], [97], [101]. A brief review of these papers can be also found in Chapter 2. We begin our investigations in Chapters 3 and 4 with the study of two types of generalized discrete convolutions referred to as the ﬁnite discrete and inﬁnite discrete case, 1 respectively. In the ﬁnite case we consider sequences{A(a)}N∈Nof matrices (thought of N N+1 operators acting onC) which are given by µ µ ¶¶ n k abn−k, , N N n,k=0,...,N where{ban(x, y)}is the sequence of the Fourier coeﬃcients of a functiona(x, y, t) deﬁned 1 on [0,1]×[0,1]×T. This function is called the symbol of the sequence{A(a)}N∈N. N
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2 In the inﬁnite case we consider sequences (a)}ch are of the {AN N∈Nof matrices whi form µ µ ¶¶ n k ban−k, , E(N)E(N) n,k=0,...,N where{ban(x, y)}is the sequence of the Fourier coeﬃcients of a functiona(x, y, t) deﬁned on [0,+∞)×[0,+∞)×Tthis function is called the symbol of the sequence. Again 2 have values in (0,+∞) and to possess {AN(a)}N∈N. The functionE(N) is assumed to the following properties:
E(N)∞→−−−, N→∞
N −−−→∞. E(N) N→∞
1 In Chapter 3 we obtain suﬃcient conditions for the asymptotic invertibility ofA(a) N asN→ ∞Using this construction, we proveand construct an almost inverse operator. the analogue of the asymptotic formula (1) ﬁrst in the case whenfis a rational function and then for a function which can be approximated by rational ones. Chapter 4 contains 2 analogous results obtained for the sequence{A(a)}N∈N. N In Chapter 5 we consider the continuous analogue of the ﬁnite discrete case referred to 1 as the ﬁnite integral case, namely the family{A(a)}τ >0of operators acting on the space τ L2([0, τ]) deﬁned by Z τ³ ´ x y 1 (A(a)f)(x) =cf(x) +, xk , −y f(y)dy, f∈L2([0, τ]), τ τ τ 0
wherecis a complex number and the integral kernelk(x, y, t) is deﬁned on [0,1]×[0,1]×R. The Fourier transform, or more precisely, the function Z iξt a(x, y, ξ) =c+e k(x, y, t)dt R
1 deﬁned on [0,1]×[0,1]×Ris called the symbol of the family{A(a)}τ >0. By analogy τ with the discrete case, we construct an almost inverse operator. We describe a complex 1 domain containing the spectrum ofA(a) forτlarge enough and obtain the asymptotics τ of the spectrum asτapproaches inﬁnity. Chapter 6 contains analogous results derived in the so-called inﬁnite integral case for 2 operators{A(a)}τ >0of the form τ Z µ ¶ τ x y 2 A(a)f)(x) = (τcf(x) +, xk , −y f(y)dy, f∈L2([0, τ]), 0E(τ)E(τ)
wherecthe functionis a complex number, k(x, y, t) is deﬁned on the set [0,+∞)×[0,+∞)×R, andE(τ) is a function having values in (0,+∞) and satisfy-ing the following conditions: τ E(τ)−∞→−−,−−−∞→. E(τ) τ→∞τ→∞
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2 ( deﬁned analogously. The symbola(x, y, ξ) of{Aτa)}τ >0is All the results mentioned above are obtained for generalized discrete and integral convolution operators, the symbols of which satisfy certain conditions. For example, the symbol in the discrete case is supposed to belong to a generalized Wiener algebra, i.e. its Fourier coeﬃcients are required to satisfy the following condition: X sup|ban(x, y)|<∞. x,y n∈N
Since not all continuous functions satisfy this condition, the next interesting problem is to weaken the conditions on the symbol. This question is considered in the present disser-∗ tation for generalized discrete convolutions by using ofC-algebra techniques developed for ordinary convolution operators by A. Böttcher, R. Hagen, S. Roch and B. Silbermann in [12], [13] and [42]. Algebraic methods applied to the generalized convolutions make it possible to widen the class of operator sequences and to describe correctly the type of operators with con-tinuous symbols. ∗1 The ﬁnite discrete case is studied in Chapter 7. We consider theC-algebraA 1 generated by operator se{Aroperties of this algebra quences of the formN(a)}N∈Np. The 1 allow to associate to each operator sequence fromAa continuous symbol and to obtain 2 the asymptotics of the spectrum. The inﬁnite discrete case, i.e. the algebraAgenerated 2 by sequences{A(a)}N∈N, is studied in Chapter 8. N We note that the direct application of these algebraic techniques to the integral case is problematic. When constructing a suitable algebraic framework for integral convolutions, we should take into account trace class properties. It does not seem to be possible to ∗ construct aC-algebra in this situation. The problem demands further investigations with more diﬃcult methods. For this reason, it is not considered here. Chapter 9 contains a summary of results obtained in the present work.
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2
General information on the theory of ordinary and generalized convolution operators
In this chapter we give the basic information on ordinary convolutions and make a review of results obtained for generalized operators.
2.1 Discrete convolution operators and Szegö limit theorems ∞ LetL(T) be the Banach space of essentially bounded functions on the unite circle T={t∈C,|t|= 1}. Letl2(Z) (Zis the set of integer numbers) be the Hilbert space of square summable sequences. ∞ Fora∈L(T), the operatorL(a) acting onl2(Z) with the matrix representation
(ban−k)n,k∈Z,
where{ban}n∈Zis the sequence of the Fourier coeﬃcients of the functiona, is called the Laurent operator with the symbola. This is a simple example of a convolution operator. It can be easily shown (see, for example, [13]) that
∞ L(a)L(b) =L(aball) for a, b∈L(T).
Therefore, the operatorL(a) is invertible if and only if its symbolaIn thisis invertible. −1−1 caseL(a) =L(a). Another important example of convolutions is the Toeplitz operatorT(a) acting on + + l2(Z) (hereZdenotes the set of nonnegative integer numbers) with the inﬁnite matrix
(abn−k)n,k=0,1,....
∞ The functiona∈L(T) is called the symbol ofT(a). The algebraic properties of Toeplitz operators are very special. following equation can be also found in [13]:
e T(a)T(b) =T(ab)−H(a)H(b)
∞ for alla, b∈L(T),
∞ whereH(a) fora∈Lis the Hankel operator with the matrix
(abn+k+1)n,k=0,1,...
The proof of the
e e ∞ −1 andbis the function fromL(T) deﬁned byb(t) =b(t) fort∈T. Thus, the problem of the invertibility for Toeplitz operators demands additional investigations and has a long history. It was investigated for various classes of generating functions. The symbols from the Wiener algebraW(that is the algebra constituted by continuous functions the Fourier series of which converge absolutely) were considered ﬁrst. Then, the corresponding result was proved for continuous symbols and subsequently for essentially bounded functions. A thorough information to this question and the proof of the next statements can be found in [13].
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∞ Proposition 2.1.1Leta∈L(T). If the convex hull of the essential range ofadoes not contain zero, then the Toeplitz operatorT(a)is invertible.
∞ Corollary 2.1.1Let the functiona∈L(T)Ifbe continuous. adoes not vanish onT and has winding number zero, then the Toeplitz operatorT(a)is invertible.
IfT(a) is invertible, we can ﬁnd a solution of the equation
T(a)x=y,
x, y∈l2
(2)
−1−1 in the formx=T(a)y. However,T(aTherefore,) is in general not explicitly known. one tries to replace equation (2) by the following truncated equation:
TN(a)xN=yN,
whereTN(a) withN∈Nhas the matrix (ban−k)n,k=0,...,N,{yN}N∈Nis the corresponding truncation of the vectory(. The N+ 1)-dimensional vectorxNis to be found. The operatorT(a) is said to be asymptotically invertible by means of the ﬁnite section method{TN(a)}N∈Nif forNlarge enough the equationsTN(a)xN=yNare uniquely solvable and the solutionxNconverges in norm to the solutionxThis is equivalentof (2). to the stability of{TN(a)}N∈N, that isTN(a) is invertible forNlarge enough, say for −1 ¯¯ ¯¯ N>N0sup, and T(a)<∞. N N>N0 The following fact is well known (see, for instance, [13]).
Proposition 2.1.2Ifais continuous, then the operatorT(a)is invertible if and only if the sequence{TN(a)}N∈Nis stable. This means,T(a)is invertible if and only ifT(a)is asymptotically invertible by means of{TN(a)}N∈N.
The sequence of truncated Toeplitz operators is often used for solving a lot of practice problems (an example, the Ising model, can be found in [13]). The asymptotic spectral properties of the truncated Toeplitz operatorsTN(a) are very important in this case. The ﬁrst results on the behavior of the spectra were obtained by G. Szegö in [99] and [100]. We adduce his theorems in the modern form.
∞ Theorem 2.1.1 (Szegö’s ﬁrst limit theorem)Leta∈L(T)and letΩ⊂Cbe an open set containing the convex hull of the essential range ofa. Iffis analytic inΩ, then Z 1 1 trf(TN(a)) =f(a(t))dµ+o(1), N→ ∞. N+ 1 2πT
This asymptotics can be precised for generating functions satisfying additional condi-½ ¾ P 1/2 ∞ ∞2 tions. Let us denoteL(T)∩B=a∈L(T() : |n|+ 1)|abn|<∞. 2 n∈Z
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