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Eigenvalues of Larger Toeplitz Matrices: the Asymptotic Approach

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Eigenvalues of Larger Toeplitz Matrices: the Asymptotic Approach Eigenvalues of larger Toeplitz matrices: the asymptotic approach. Sergei M. Grudsky (CINVESTAV, Mexico-City, Mexico) Moscow, October 2010 Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 1 / 148 Main object. Spectral properties of larger finite Toeplitz matrices a0 a−1 a−2 ... a−(n−1) a a a ... a 1 0 −1 −(n−2) n−1 An = (a ) = a2 a1 a0 ... a−(n−3) . j−k j,k=0 . .. . an−1 an−2 an−3 ... a0 Eigenvalues, singular values, condition numbers, invertibility and norms of inverses, e.t.c. n ∼ 1000 is a business of numerical linear algebra. Statistical physics - n = 107 − 1012 - is a business of asymptotic theory. Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 2 / 148 Limit matrix. a0 a−1 a−2 ... ∞ a1 a0 a−1 ... A = (aj−k )j,k=0 = a2 a1 a0 ... ............ Question Does spectral properties of A is a limit (in some sense) of {An} or not? Yes: - properties invertibility and norms of inverses (for a larger class of symbols); - limiting set in a case of real-value symbols. No: - distribution of eigenvalue in general (complex-value) case. Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 3 / 148 Content of the Course. 1. Infinite Toeplitz matrices. 2. Finite section method, stability. 3. Szeg¨o’s limit theorems. 4. Limiting spectral set of sequences of Toeplitz matrices. 5. Asymptotics of eigenvalues. Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 4 / 148 Literature 1.A.B ¨ottcher, S. Grudsky.Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis. Hindustan Book Agency, New Delhi, 2000 and Birkh¨auser Verlag, Basel,Boston, Berlin, 2000. 2.A.B ¨ottcher, B. Silbermann. Introduction to Large Truncated Toeplitz Matrices. Universitext, Springer-Verlag, New York, 1999. 3. Albrecht B¨ottcher and Sergei M. Grudsky. Spectral Properties of Banded Toeplitz Matrices. SIAM, Philadelphia, 2005, pp. 411. 4. Albrecht B¨ottcher, Sergei M. Grudsky and Egor A. Maksimenko. Pushing the envelope of the test functions in the Szeg¨o and Avram-Parter theorems. Linea Algebra and its Applications 429 (2008), 346-366. Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 5 / 148 Literature 5. Albrecht B¨ottcher, Sergei M. Grudsky, Egor A. Maksimenko. On the structure of the eigenvectors of large Hermitian Toeplitz band matrices. Operator Theory: Advances and Applications 210 (2010), 15-36. 6. Albrecht B¨ottcher, Sergei M. Grudsky, Egor A. Maksimenko. Inside the eigenvalues of certain Hermitian Toeplitz band matrices. J. of Computational and Applied Mathematics 233 (2010), 2245-2264. Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 6 / 148 Infinite Toeplitz Matrices Boundedness ∞ Given a sequence {an}n=−∞ of complex numbers, an ∈ C, when does the matrix a0 a−1 a−2 ... a1 a0 a−1 ... A = (1) a2 a1 a0 ... ............ 2 2 induce a bounded operator on l := l (Z+), where Z+ is the set of nonnegative integers, Z+ := {0, 1, 2,...}? The answer is classical result by Otto Toeplitz. Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 7 / 148 Theorem (Toeplitz 1911) The matrix (1) defines a bounded operator on l2 if and only if the ∞ numbers {an} are the Fourier coefficients of some function a ∈ L (T), Z 2π 1 iθ −inθ an = a(e )e dθ, n ∈ Z. (2) 2π 0 In that case the norm of the operator given by (1) equals kak∞ := ess sup |a(t)|. t∈T Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 8 / 148 Proof. We denote by L2 := L2(T) and L∞ := L∞(T) the usual Lebesgue spaces on the complex unit circle T. The multiplication operator M(a): L2 → L2, f 7→ af ∞ is bounded if and only if a is in L , in which case kM(a)k = kak∞. An 2 ∞ orthonormal basis of L is given by {en}n=−∞ where 1 n en(t) = √ t , t ∈ T. 2π Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 9 / 148 The matrix representation of M(a) with respect to the basis {en} is easily seen to be ......... ............ ... a0 a−1 a−2 a−3 a−4 ... ... a1 a0 a−1 a−2 a−3 ... L(a) := ... a a a a a ... (3) 2 1 0 −1 −2 ... a a a a a ... 3 2 1 0 −1 ... a4 a3 a2 a1 a0 ... ......... ............ where the an’s are defined by (2). Thus, we arrive at the conclusion that L(a) defines a bounded operator on l2(Z) if and only if a ∈ L∞ and that kL(a)k = kak∞ in this case. The matrix (1) is the lower right quarter of L(a), that is we may think of 2 2 A as the compression of L(a) to the space l = l (Z+). This implies that if a ∈ L∞, then kAk ≤ kL(a)k = kak∞. (4) Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 10 / 148 2 For a natural number n, let Sn be the projection on l (Z) given by ∞ Sn :(xk )k=−∞ 7→ (..., 0, 0, x−n,..., x−1, x0, x1, x2,...). The matrix representation of the operator SnL(a)Sn|Im Sn results from (3) by deleting all rows and columns indexed by a number in {−(n + 1), −(n + 2),...}. Hence, SnL(a)Sn|Im Sn has the matrix (1) as its matrix representation. This shows that kAk = kSnL(a)Snk. (5) Because Sn converges strongly (=pointwise) to the indentity operator on 2 l (Z), it follows that SnL(a)Sn → L(a) strongly, whence kL(a)k ≤ lim inf kSnL(a)Snk. (6) n→∞ Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 11 / 148 From (5) and (6) we see that L(a) and thus M(a) must be bounded whenever A is bounded and that kL(a)k ≤ kAk. (7) Consequently, A is bounded if and only if a ∈ L∞, in which case (4) and (7) give the equality kAk = kak∞. Clearly, if there is a function a ∈ L∞ satisfying (2), then this function (or, to be more precise, the equivalence class of L∞ containing it) is unique. We therefore denote both the matrix (1) and the operator it induces on l2 by T (a). The function a is in this context referred to as the symbol of the Toeplitz matrix/operator T (a). Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 12 / 148 Compactness and Selfadjointness In this section we cite two very simple but instructive results. They reveal that Toeplitz operators with properly complex-valued symbol cannot be tackled by the tools available for compact and selfadjoint operators. Proposition (Gohberg 1952) The only compact Toeplitz operator is the zero operator. ∞ Proof. Let a ∈ L and suppose T (a) is compact. Let Qn be the projection 2 2 Qn : l → l , (x0, x1, x2,...) 7→ (0,..., 0, xn, xn+1,...). (8) As Qn → 0 strongly and T (a) is compact, it follows that kQnT (a)Qnk converges strongly to 0. But the compression QnT (a)Qn|Im Qn has the same matrix as T (a) whence kT (a)k = kQnT (a)Qnk. Consequently, T (a) = 0. Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 13 / 148 Because T (a) − λI = T (a − λ) for every λ ∈ C, we learn from Proposition 2 that T (a) is never of the form λI+ a compact operator unless T (a) = λI. Proposition The Toeplitz operator T (a) is selfadjoint if and only if a is real-valued. Proof. This is obvious: T (a) is selfadjoint if and only if an = a−n for all n, which happens if and only if a(t) = a(t) for all t ∈ T. Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 14 / 148 C∗-Algebras A Banach algebra is a Banach space A with an associative and distributive multiplication such that kabk ≤ kak kbk for all a, b, ∈ A. If a Banach algebra A has a unit element, which is usually denoted by e, 1, or I, it is referred to as a unital Banach algebra. A conjugate-linear map a 7→ a∗ of a Banach algebra into itself is called an involution if a∗∗ = a and (ab)∗ = b∗a∗ for all a, b ∈ A. Finally, a C ∗-algebra is a Banach algebra with an involution such that ka∗ak = kak2 for all a ∈ A. In more detail, we can define a C ∗-algebra as follows. A C ∗-algebra is a set A with four algebraic operations and a norm. The four algebraic operations are multiplication by scalars in C, addition, multiplication, and involution. The following axioms must be satisfied for the operations: Sergei M. Grudsky (CINVESTAV,Mexico) Eigenvalues of lager Toeplitz matrices Moscow, October 2010. 15 / 148 (1) the axioms of a linear space for scalar multiplication and addition; (2) a(bc) = (ab)c for all a, b, c ∈ A; (3) a(b + c) = ab + ac, (a + b)c = ac + bc for all a, b, c ∈ A; (4) (λa)∗ = λa∗, (a + b)∗ = a∗ + b∗, (ab)∗ = b∗a∗ for all λ ∈ C and a, b ∈ A. The norm is subject to the following axioms: (5) the axioms of a normed space; (6) A is complete (that is, a Banach space); (7) kabk ≤ kak kbk for all a, b ∈ A; (8) ka∗ak = kak2 for all a ∈ A.
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