Vol. 103, No. 2, 1982

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Estimates of meromorphic functions and summability theorems

Andrei Shkalikov

Vol. 103 (1982), No. 2, 569–582
Abstract

The main goal of this paper is to prove the following theorem.

Theorem 1. Let L be an unbounded operator in a Hilbert space H, having a discrete spectrum {λj}⊂ G = BR Pq,h, where BR = {λ : |λ|R}, Pq,h = {λ : Re λ 0,|λ| > 1,|Im λ|h(Re λ)q,h > 0,−∞ < q < 1}, and for some γ < , L1 σγ. Also let the estimate

        −1      −1         --
∥(Iλ − L)  ∥ ≦ Cd  (λ,G),  λ∈G

hold outside the domain G= BR Pq,2h, and for some a > 0, p > 0

 ∑   1 = n(t) ≦ dtp
|λj|≦t

provided t is sufficiently large.

Then L A(α,H) for any α > max0, p (1 q).

Besides, if the numbers a or h can be chosen arbitrarily small and p(1 q) > 0, then α = p (1 q) is admissible.

Mathematical Subject Classification 2000
Primary: 47A70
Secondary: 30D30
Milestones
Received: 14 April 1980
Published: 1 December 1982
Authors
Andrei Shkalikov