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In this paper we use Riesz spectral Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of pseudo-differential operators defined on the unit circle $\mathbb{T} := \mathbb{R}/ 2 \pi \mathbb{ Z}$. For symbols…

Let A be a self-adjoint operator on a separable Hilbert space H. Assume that the spectrum of A consists of two disjoint components s_0 and s_1 such that the set s_0 lies in a finite gap of the set s_1. Let V be a bounded self-adjoint…

Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a bounded normal right quaternionic linear operator on $\mathcal{H}$. In this paper, we prove that there exists a unique spectral measure $E$ in $\mathcal{H}$ such that…

If $A$ is a compact operator in a Banach space and some power $A^q$ is nuclear we give a criterion of $\mathbb{Z}_{d}$ -- symmetry of its spectrum $\sigma{A}$ in terms of vanishing of the traces $\mathop{\mathit{Trace}} A^n$ for all $n$, $n…

Let $T:A\to X$ be a bounded linear operator, where $A$ is a C$^*$-algebra, and $X$ denotes an essential Banach $A$-bimodule. We prove that the following statements are equivalent: $(a)$ $T$ is anti-derivable at zero (i.e. $ab =0$ in $A$…

A complete analysis of the essential spectrum of matrix-differential operators $\mathcal A$ of the form \begin{align} \begin{pmatrix} -\displaystyle{\frac{\rm d}{\rm d t}} p \displaystyle{\frac{\rm d}{\rm d t}} + q &…

We present an operator-coefficient version of Sato's infinite-dimensional Grassmann manifold, and tau-function. In this context, the Burchnall-Chaundy ring of commuting differential operators becomes a C*-algebra, to which we apply the…

In an attempt to propose more general conditions for decoherence to occur, we study spectral and ergodic properties of unital, completely positive maps on not necessarily unital $C^*$-algebras, with a particular focus on gapped maps for…

We show that the following conditions on a C*-algebra are equivalent: (i) it has the fixed point property for nonexpansive mappings, (ii) the spectrum of every self adjoint element is finite, (iii) it is finite dimensional. We prove that…

We prove spectral multiplier theorems for H\"ormander classes $\mathcal{H}^\alpha\_p$ for 0-sectorial operators A on Banach spaces assuming a bounded $H^\infty(\Sigma\_\sigma)$ calculus for some $\sigma \in (0,\pi)$ and norm and certain…

We present results on the broadband nature of the power spectrum $S(\omega)$, $\omega\in(0,2\pi)$, for a large class of nonuniformly expanding maps with summable and nonsummable decay of correlations. In particular, we consider a class of…

In this article, we prove the following spectral theorem for right linear normal operators (need not to be bounded) in quaternionic Hilbert spaces: Let $T$ be an unbounded right quaternionic linear normal operator in a quaternionic Hilbert…

Motivated by Cuntz-Krieger-Toeplitz systems associated to undirected graphs and representations of groupoids, we obtain a generalisation of the Sz-Nagy's Dilation Theorem for operator valued partially positive semidefinite maps on…

Let $S=\{p_1, \dots, p_r,\infty\}$ for prime integers $p_1, \dots, p_r.$ Let $X$ be an $S$-adic compact nilmanifold, equipped with the unique translation invariant probability measure $\mu.$ We characterize the countable groups $\Gamma$ of…

It is shown that if $C$ is a nonempty convex and weakly compact subset of a Banach space $X$ with $M(X)>1$ and $T:C\rightarrow C$ satisfies condition $(C)$ or is continuous and satisfies condition $(C_{\lambda})$ for some $\lambda \in…

Let $B(H)$ be the algebra of all bounded linear operators on infinite-dimensional complex Hilbert space $H$. For $T, S \in B(H)$ denote by $T\bullet S=TS+ST^{\ast}$ and $[T\circ S]_{\ast}=TS-ST^{\ast}$ the Jordan $\ast$-product and the skew…

For a tuple $A=(A_0, A_1, ..., A_n)$ of elements in a unital Banach algebra ${\mathcal B}$, its {\em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0, z_1, ..., z_n]\in \pn$ such that $A(z)=z_0A_0+z_1A_1+... +z_nA_n$…

We investigate the spectrum of the generator induced on the space of Hilbert-Schmidt operators by a Gaussian quantum Markov semigroup with a faithful normal invariant state in the general case, without any symmetry or quantum detailed…

Let $A$ be a $C^{*}$ algebra and $T: A\rightarrow A$ be a linear map which satisfies the functional equation $\begin{cases}T(x)T(y)=T^{2}(xy)\\T(x^{*})=T(x)^{*} \end{cases}$ We prove that under each of the following conditions, $T$ must be…

Let $E$ be a real Banach space. We study the Ornstein-Uhlenbeck semigroup $P(t)$ associated with the Ornstein-Uhlenbeck operator $$ Lf(x) = \frac12 {\rm Tr} Q D^2 f(x) + <Ax, Df(x)>.$$ Here $Q$ is a positive symmetric operator from $E^*$ to…