Spectral Inclusion and Analytic Continuation (original) (raw)
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For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These results are used to prove sufficient conditions for eigenvalue accumulation to the poles and to infinity of rational operator functions. Finally, an application of electromagnetic field theory is given.
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Journal of Mathematical Analysis and Applications
In this paper we study the behavior of holomorphic mappings on A-compact sets. Motivated by the recent work of Aron, Ç alişkan, García and Maestre (2016), we give several conditions (on the holomorphic mappings and on the λ-Banach operator ideal A) under which A-compact sets are preserved. Appealing to the notion of tensorstability for operator ideals, we first address the question in the polynomial setting. Then, we define a radius of (A; B)compactification that permits us to tackle the analytic case. Our approach, for instance, allows us to show that the image of any (p, r)-compact set under any holomorphic function (defined on any open set of a Banach space), is again (p, r)-compact.
Mathematical Surveys and Monographs, 1980
As usual we let R be any discrete subring of the complex numbers. Throughout this chapter R will always contain the identity. DEFINITION 11.1. Let/be a complex valued function on a subset X of C. For each n in N set £"(/)= inf \\f-p\\ x deg/></i where the/?'s have arbitrary complex coefficients and £"<(/)= inf ||/-9H* deg q < n where the coefficients of the q's he in R. Clearly E Q {f) > E x (f) > , ES(f) > W) > , 0) and E"(f) < E< n (f), n e N. (2) In Part II we were concerned with characterizing those/for which E*(f)->0 as n-» oo. In Part III we will obtain information about the asymptotic behavior of the sequence {E*(f)} as n-> oo. From (2) we have a sort of lower bound on the rate of convergence of E*(f) to zero which we will not mention again. In the present chapter we will be concerned with the effect on the sequence {E*(f)} of the hypothesis that/be analytic. If/is analytic on an interval, then by a well-known theorem of S. N. Bernstein (Lorentz [66, 5.5]) there exists p < 1 such that E n (f) = 0(p n), i.e., there exists a constant C such that E n (f) < Cp n , all n in N. A similar result holds for E%(f). We start with the following. THEOREM 11.2. Let q be a monic polynomial in R[z], 0 < p, < p 2 , and z v. .. , z k the zeros of q. If f is analytic where \q{z)\ < p 2 and X = (z: \q(z)\ < pj}, then E£(f) = 0(p n) on X for some p < \ if and only if for every positive integer r there is a polynomial q r in R [z] satisfying tf\zj) = f (*\zj), \<j<k,0<s<r.
On bounded analytic functions in finitely connected domains
1987
A new proof of the corona theorem for finitely connected domains is given. It is based on a result on the existence of a meromorphic selection from an analytic set-valued function. The latter fact is also applied to the study of finitely generated ideals of H°° over multiply connected domains. Introduction. In this paper, which is a direct continuation of [18], we study applications of analytic multifunctions to some topics in function theory on finitely connected domains, related mainly to the corona problem. Concerniag analytic multifunctions (which are certain set-valued functions, cf. Definition 1.2), §§1-3 of [18] form sufficient background for our purposes here. The reader is also referred there for the information about the origin of the approach employed in this paper. However, basic definitions and references are provided below. Our basic new result (Theorem 1.4) associates to every analytic multifunction defined in a finitely connected domain some meromorphic functions wit...
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Monatshefte Fur Mathematik, 1988
LetE be a compact subset of the complex planeC such that Leja's extremal functionL E forE is continuous. If almost all zeros of the polynomials of best approximation to a functionf∈C(E) are outside the setE R ={z∈C:L E (z<R)}, for someR>1, thenf is extendible to a holomorphic function inE R . If the zeros ofn-th, polynomial of best approximation tof are outside ERnE_{R_n } ERn and the sequence {R n −n } rapidly decreases to zero thenf can be extended to aC ∞ function on 075-4}.
C2 extensions of analytic functions defined in the complex plane
Advances in Applied Clifford Algebras, 2001
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Extensions of harmonic and analytic functions
Pacific Journal of Mathematics, 1990
This paper studies the extensions of harmonic and analytic functions defined on the unit disk to continuous functions defined on a certain compactification of the disk.