Related papers: A Generalized Spectral Radius Formula and Olsen's …

Let $\sigma(A)$, $\rho(A)$ and $r(A)$ denote the spectrum, spectral radius and numerical radius of a bounded linear operator $A$ on a Hilbert space $H$, respectively. We show that a linear operator $A$ satisfying $$\rho(AB)\le r(A)r(B)…

Given the norms of powers $(\lVert x^n\rVert)_{n\geq 0}$ of a Banach algebra element $x$, the largest possible value of the minimum modulus on the spectrum of $x$ is determined. It is also shown that, given a Banach algebra element $x$ and…

Let $r_A(T)$ denote the $A$-spectral radius of an operator $T$ which is bounded with respect to the seminorm induced by a positive operator $A$ on a complex Hilbert space $\mathcal{H}$. In this paper, we aim to establish some $A$-spectral…

We study the spectrum of operators $aT\in B(H)$ on a Hilbert space $H$ where $T$ is an isometry and $a$ belongs to a commutative $C^*$-subalgebra $C(X)\cong A\subseteq B(H)$ such that the formula $L(a)=T^*aT$ defines a faithful transfer…

For an arbitrary commuting $d$--tuple $\bT$ of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform $\dbT$ and we prove that the spectral radius of $\bT$ can be calculated from the…

We study different operator radii of homomorphisms from an operator algebra into $B(H)$ and show that these can be computed explicitly in terms of the usual norm. As an application, we show that if $\Omega$ is a $K$-spectral set for a…

Operators possessing analytic generalized inverses satisfying the resolvent identity are studied. Several characterizations and necessary conditions are obtained. The maximal radius of regularity for a Fredholm operator T is computed in…

Suppose that $\lambda - T$ is left-invertible in $L(H)$ for all $\lambda \in \Omega$, where $\Omega$ is an open subset of the complex plane. Then an operator-valued function $L(\lambda)$ is a left resolvent of $T$ in $\Omega$ if and only if…

Using ergodic theory, in this paper we present a Gel'fand-type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup $\bS^+$ restricted to a…

In this short note we first extend the validity of the spectral radius formula obtained in \cite{ag} to Fourier--Stieltjes algebras. The second part is devoted to showing that for the measure algebra on any locally compact non-discrete…

For any $r$-order $\{0, 1\}$-tensor $A$ with $e$ ones, we prove that the spectral radius of $A$ is at most $e^{\frac{r-1}{r}}$ with the equality holds if and only if $e={k^r}$ for some integer $k$ and all ones forms a principal sub-tensor…

In this paper, we prove the relation $\frac{r_{A}(T) + r_{A}(T^{\diamond}) + |r_{A}(T^{\diamond}) - r_{A}(T)|}{2} = \sup \{ |\lambda|: \lambda \in \sigma_{A}(T)\}$, where $A$ is a positive semidefinite operator (not necessarily to have a…

In the classical operator theory, there are several versions of spectra, related to special classes of operators (Fredholm, semi-Fredholm, upper/lower semi-Fredholm,etc.). We generalize these notions for adjointable operators on Hilbert…

Given a Hilbert module $H$ over a $C^*$-algebra, let $\mathcal{L}(H)$ be the set of all adjointable operators on $H$. For each $T\in\mathcal{L}(H)$, its numerical radius is defined by $w(T)=\sup\big\{\|\langle Tx, x \rangle\|: x\in H,…

We prove that for any two elements $A$, $B$ in a factor $M$, if $B$ commutes with all the unitary conjugates of $A$, then either $A$ or $B$ is in $\mathbb{C}I$. Then we obtain an equivalent condition for the situation that the $C$-numerical…

In this paper, we shall discuss topics in geometry of numbers in the positive characteristic setting, such as covering radii. We find a closed form for covering radii with respect to convex bodies, which will lead to a proof of the positive…

In this paper, a spectral theorem is proved for self-adjoint cyclically compact partial integral operators in the space of functions with mixed norm, which is a Kaplansky--Hilbert module. The decomposition through eigenfunctions, integral…

Let $(X,+,d)$ be an Abelian metric group and $A\subset X$. We investigate the spectre of a set $A$, defined as the set of all elements $z\in X$ such that for every $x\in A$ either $x+z \in A$ or $x-z \in A$. We consider the corresponding to…

A generalized Krein-Rutman theorem for a strongly positive bounded linear operator whose spectral radius is larger than essential spectral radius is established: the spectral radius of the operator is an algebraically simple eigenvalue with…

Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in…