# Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces, II

Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces, II We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator $$\mathscr {L}= -\mathrm{d}/\mathrm{d}t+A$$ L = - d / d t + 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 invertibility implies a stronger state for $$\mathscr {L}$$ L under mild additional conditions. For example, we show that if the operator $$\mathscr {L}$$ L is surjective and the imaginary axis is not contained in the interior of the spectrum of A, then $$\mathscr {L}$$ L is invertible. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

# Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces, II

, Volume 14 (4) – Aug 2, 2017
13 pages

/lp/springer_journal/spectral-analysis-of-abstract-parabolic-operators-in-homogeneous-wnYQTqB0Hz
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-017-0982-y
Publisher site
See Article on Publisher Site

### Abstract

We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator $$\mathscr {L}= -\mathrm{d}/\mathrm{d}t+A$$ L = - d / d t + 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 invertibility implies a stronger state for $$\mathscr {L}$$ L under mild additional conditions. For example, we show that if the operator $$\mathscr {L}$$ L is surjective and the imaginary axis is not contained in the interior of the spectrum of A, then $$\mathscr {L}$$ L is invertible.

### Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Aug 2, 2017

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