By Albrecht Böttcher, Bernd Silbermann, Alexei Yurjevich Karlovich

A revised creation to the complex research of block Toeplitz operators together with fresh study. This booklet builds at the good fortune of the 1st version which has been used as a customary reference for fifteen years. subject matters variety from the research of in the community sectorial matrix services to Toeplitz and Wiener-Hopf determinants. it will entice either graduate scholars and experts within the idea of Toeplitz operators.

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14 were ﬁrst obtained by N. Ya. Krupnik (see Kesler and Krupnik [311]). 14 itself was ﬁrst established by K¨ ohler and Silbermann [314], [315] and independently also by Markus and Feldman [347]. 15 is due to Markus and Feldman [346]. The proofs of these two theorems are in the original papers and also in Chapter 1 of Krupnik’s book [329]. (We also recommend to have a glance at Math. Reviews 87a:15006). 29. This material is well known and a major part of it is in almost each ˙ [587], [588]. book on Banach or C ∗ -algebras.

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.

6) is called the Toeplitz operator on p generated by the function a. 6) is usually referred to as the symbol of the corresponding operator. Toeplitz operators on p are sometimes also called discrete Wiener-Hopf operators. 6), respectively, are unitarily equivalent through the isomorphism H2 → 2 ϕn χn → {ϕn }n∈Z+ . 7) n∈Z+ Therefore we shall frequently identify these operators without mentioning this explicitly. For f ∈ H p , g ∈ H q , ϕ ∈ p , ψ ∈ q (1/p + 1/q = 1) let (f, g) := 1 2π f g dm, (ϕ, ψ) := T ϕn ψn .