Mediterr. J. Math.
Springer International Publishing AG 2017
Spectral Analysis of Abstract Parabolic
Operators in Homogeneous Function
Anatoly G. Baskakov and Ilya A. Krishtal
Abstract. We use methods of harmonic analysis and group representa-
tion theory to study the spectral properties of the abstract parabolic
operator L = −d/dt + A in homogeneous function spaces. We focus on
the dependency between various invertibility states of such an operator.
In particular, we prove that often, a generally weaker state of invertibil-
ity implies a stronger state for L under mild additional conditions. For
example, we show that if the operator L is surjective and the imagi-
nary axis is not contained in the interior of the spectrum of A,thenL
Mathematics Subject Classiﬁcation. 47A10, 46H25.
Keywords. Abstract parabolic operators, homogeneous function spaces,
Beurling spectrum, invertibility states.
In this paper, we continue our study of the spectral properties of a diﬀerential
L = −d/dt + A : D(L ) ⊂F(R,X) →F(R,X) (1.1)
in homogeneous Banach spaces F(R,X) of functions with values in a complex
Banach space X. The operator A : D(A) ⊂ X → X in (1.1) is assumed to
be the inﬁnitesimal generator of a C
-semigroup T : R
=[0, ∞) → B(X).
The homogeneous spaces F = F(R,X) and the operator L are identiﬁed
precisely in Deﬁnitions 2.1 and 2.6, respectively.
This paper is a sequel to , where the focus is on the basic spec-
tral properties of L as well as its special properties in the space of functions
with absolutely summable spectrum. Here, the emphasis is on the interdepen-
dency between various states of invertibility of the operator L (see [6,7]and
A. G. Baskakov is supported in part by the Ministry of Education and Science of the
Russian Federation in the frameworks of the project part of the state work quota (Project
no. 1.3464.2017/4.6). I. A. Krishtal is supported in part by NSF Grant DMS-1322127.