By Chuang C.T. (ed.)

ISBN-10: 0821850504

ISBN-13: 9780821850503

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**Additional info for Analytic Functions of One Complex Variable**

**Example text**

Evidently, for all n > 2, k Cn−1 = 2 γn,k = (n − 1)(n − 2) . . , n − 1). k (n − 1) k! k! 1 Let A be a linear operator in Cn . Then its resolvent satisfies the inequality n−1 Rλ (A) ≤ k=0 g k (A)γn,k for any regular point λ of A, ρk+1 (A, λ) where ρ(A, λ) = min |λ − λk (A)|. ,n That is, ρ(A, λ) is the distance between σ(A) and a point λ ∈ C. 6. 1 is exact: if A is a normal matrix, then g(A) = 0 and Rλ (A) = 1 for all regular points λ of A. 2 Let A be a linear operator in Cn . Then n−1 √ Rλ (A) ≤ k=0 g k (A) for any regular point λ of A.

CLASSES OF OPERATORS In addition, let A be of the Hille-Tamarkin type [Lp , Lq ]. Then the eigenvalues of A satisfy the inequality ∞ |λk (A)|ν < ∞ k=1 with ν = max {2, p}. This result is best possible. 7 Let 1 1 + ≤ 1. p q In addition, let A be of the Hille-Tamarkin type [Lp , Lq ]. Then the eigenvalues of A satisfy the inequality ∞ |λk (A)|q+ < ∞ k=1 with q+ = max {2, q } where 1 1 + = 1. q q This result is best possible. Chapter 3 Functions of Finite Matrices Throughout the present chapter B(Cn ) is the set of all linear operators in the Euclidean space Cn (matrices), .

If there is a constant a, such that the inequality Ah Y ≤a h X for all h ∈ X holds, then the operator is said to be bounded. The quantity A X→Y := sup h∈X Ah Y h X is called the norm of A. If X = Y we will write A X→X = A X or simple A . A linear operator is said to be completely continuous (compact) if it is bounded and maps each bounded set in X into a compact one in Y . 3. LINEAR OPERATORS 5 Under the natural definitions of addition and multiplication by a scalar, and the norm, the set B(X, Y ) of all bounded linear operators acting from X into Y becomes a Banach space.

### Analytic Functions of One Complex Variable by Chuang C.T. (ed.)

by Donald

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