Spectral Mapping Theorem for C0-Semigroups of Drazin spectrum

Let $(T(t))_{t\geq 0}$ be a $C_0$ semigroup of bounded linear operators on a Banach space $X$ and denote its generator by $A$. A fundamental problem to decide whether the Drazin spectrum of each operator $T(t)$ can be obtained from the Drazin spectrum of $A$. In particular, one hopes that the Drazin Spectral Mapping Theorem holds, i.e., $e^{t \sigma_{D}(A)}=\sigma_{D}(T(t))\backslash \{0\}$ for all $t \geq 0$.

On properties of the operator equation TT*=T+T*

In this paper, we study properties of the operator equation TT*=T+T* which T.T. West observed in [12]. We first investigate the structure of solutions T 2 B(H) of such equation. Moreover, we prove that if T is a polynomial root of solutions of that operator equation, then the spectral mapping theorem holds for Weyl and essential approximate point spectra of T and f(T) satisfies a-Weyl?s theorem for f?H(?(T)), where H(?(T)) is the space of functions analytic in an open neighborhood of ?(T).

Quaternionic quantum walks of Szegedy type and zeta functions of graphs

We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.

Ascent, essential ascent, descent and essential descent for a linear relation in a linear space

For a linear relation in a linear space some spectra defined by means of ascent, essential ascent, descent and essential descent are introduced and studied. We prove that the algebraic ascent, essential ascent, descent and essential descent spectrum of a linear relation in a linear space satisfy the polynomial spectral mapping theorem. As an application of the obtained results we show that the topological ascent, essential ascent, descent and essential descent spectrum verify the polynomial spectral mapping theorem.