By Albrecht Böttcher, Bernd Silbermann, Alexei Yurjevich Karlovich

A revised creation to the complicated research of block Toeplitz operators together with fresh examine. This publication builds at the luck of the 1st variation which has been used as a regular reference for fifteen years. themes variety from the research of in the community sectorial matrix services to Toeplitz and Wiener-Hopf determinants. it will attract either graduate scholars and experts within the thought of Toeplitz operators.

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**Example text**

N }) the operator in L(CN ) which is deﬁned by Iij : (zn , . . , zN ) → (0, . . , 0, zj , 0, . . , 0) where the zj occupies the i-th place. (b) Let A be a commutative C ∗ -algebra. Then there exists exactly one C ∗ norm in AN ×N . This norm can be given by N a AN ×N = sup aij ϕ(Iij ) i,j=1 A : ϕ ∈ (L(CN ))∗ , ϕ = 1 , where a = (aij )N i,j=1 . Proof. 5]. Note that the linear space AN ×N can be naturally identiﬁed with the algebraic tensor product A ⊗ L(CN ). The above reference implies that there is precisely one C ∗ -norm in A ⊗ L(CN ), namely, the norm which generates the injective tensor product.

N=0 n=−N A famous theorem of M. Riesz states that cp := sup f ∈P,f =0 Pf p f p is ﬁnite if 1 < p < ∞ and inﬁnite if p = 1 or p = ∞. Thus, if 1 < p < ∞ then P extends from the dense subset P of Lp to a bounded projection of the whole Lp onto H p . In 2000, Hollenbeck and Verbitsky [286] proved that cp = 1/ sin(π/ max{p, q}) where 1/p + 1/q = 1. ◦ p In the case p = 2, P is the orthogonal projection of L2 onto H 2 . Let H− p denote the kernel of P and put Q = I − P . Then L decomposes into the · ◦ p and direct sum H p +H− H p = Im P = Ker Q, ◦ p H− = Im Q = Ker P.

Basic properties of C ∗ -algebras. (a) (Gelfand/Naimark). If A is a commutative C ∗ -algebra with identity, then the Gelfand map is an isometric star-isomorphism of A onto C(M (A)). (b) (Gelfand/Naimark). If A is any C ∗ -algebra, then there exists a Hilbert space H such that A is isometrically star-isomorphic to some C ∗ subalgebra of L(H). (c) If A is a commutative C ∗ -algebra with identity, then ∂S M (A) = M (A). (d) If A is a C ∗ -algebra with identity and B is a C ∗ -subalgebra of A containing the identity, then an element a ∈ B is left (right, resp.