Complex Anal. Oper. Theory
and Operator Theory
Frame of Exponentials Related to Analytic Families
Operators and Application to a Non-self Adjoint
Problem of Radiation of a Vibrating Structure in a
· Hanen Ellouz
Received: 9 January 2018 / Accepted: 26 May 2018
© Springer International Publishing AG, part of Springer Nature 2018
Abstract In the present paper, we investigate under sufﬁcient conditions the existence
of frames of exponential families, where the exponents coincide with the eigenvalues
of the perturbed operator
T (ε) := T
+··· ,ε∈ C.
is a closed densely deﬁned linear operator on a separable Hilbert space H with
) having isolated eigenvalues with multiplicity one and T
linear operators on H having the same domain D ⊃ D(T
) and satisfying a speciﬁc
growing inequality. The obtained results are applied to a non-self adjoint problem
deduced from a perturbation method for sound radiation.
Keywords Frames of exponentials · Eigenvalues · Elastic membrane · Integro-
Communicated by Daniel Aron Alpay.
National School of Electronics and Telecommunications of Sfax, BP 1163, CP 3018 Sfax, Tunisia
Department of Mathematics, Faculty of Sciences of Sfax, BP 1171, 3000 Sfax, Tunisia