We show that the essential spectral radius ${\varrho }_e\left(T\right)$ of T ∈ B(H) can be calculated by the formula ${\varrho }_e\left(T\right)$ = inf${ℱ}_{♯·♯}\left(XT{X}^{-1}\right)$: X an invertible operator, where ${ℱ}_{♯·♯}\left(T\right)$ is a Φ₁-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if ${}_{♯·♯}\left(T\right)$ is a Φ₂-perturbation function [loc. cit.] and if T is a Fredholm operator, then $dist(0,{\sigma }_e\left(T\right))$ = sup${}_{♯·♯}\left(XT{X}^{-1}\right)$: X an invertible operator.

Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator $V_{\varphi }:f\left(x\right)\mapsto {\int }_0^{\varphi \left(x\right)}f\left(t\right)dt$ be defined on L₂[0,1]. We prove that $V_{\varphi }$ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator $V_{\varphi }$ always equals 1.

We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties $\left(\mathrm{SBaw}\right)$, $\left(\mathrm{SBab}\right)$, $\left(\mathrm{SBw}\right)$ and $\left(\mathrm{SBb}\right)$ are not preserved under direct sums of operators. However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $\left(\mathrm{SBab}\right)$, then $S\oplus T$ has the property $\left(\mathrm{SBab}\right)$ if and only if ${\sigma }_{{\mathrm{SBF}}_{+}^{-}}(S\oplus T)={\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(S\right)\cup {\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(T\right)$, where ${\sigma }_{{\mathrm{SBF}}_{+}^{-}}\left(T\right)$ is the upper semi-B-Weyl spectrum of $T$. We obtain analogous preservation results for the properties $\left(\mathrm{SBaw}\right)$,...

An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = |A|: A ⊂ X and A → p the convergence spectrum of p in X and cS(X) = ⋃cS(x,X): x ∈ X the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = χ(p,Y): p is non-isolated in Y ⊂ X, and χS(X) = ⋃χS(x,X): x ∈ X is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then κ,cf(κ) ⊂ cS(p,X). A selection of our...

We consider simultaneous solutions of operator Sylvester equations $A_{i}X-X{B}_i=C_i$ (1 ≤ i ≤ k), where $(A₁,...,A_k)$ and $(B₁,...,B_k)$ are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and $(C₁,...,C_k)$ is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of $(A₁,...,A_k)$ and $(B₁,...,B_k)$ do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

For a linear operator T in a Banach space let ${\sigma }_p\left(T\right)$ denote the point spectrum of T, let ${\sigma }_{p,n}\left(T\right)$ for finite n > 0 be the set of all $\lambda \in {\sigma }_p\left(T\right)$ such that dim ker(T - λ) = n and let ${\sigma }_{p,\infty }\left(T\right)$ be the set of all $\lambda \in {\sigma }_p\left(T\right)$ for which ker(T - λ) is infinite-dimensional. It is shown that ${\sigma }_p\left(T\right)$ is ${ℱ}_{\sigma }$, ${\sigma }_{p,\infty }\left(T\right)$ is ${ℱ}_{\sigma \delta }$ and for each finite n the set ${\sigma }_{p,n}\left(T\right)$ is the intersection of an ${ℱ}_{\sigma }$ set and a ${}_{\delta }$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

First, some classic properties of a weighted Frobenius-Perron operator ${𝒫}_{\varphi }^u$ on $L^1\left(\Sigma \right)$ as a predual of weighted Koopman operator $W=u{U}_{\varphi }$ on $L^{\infty }\left(\Sigma \right)$ will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of ${𝒫}_{\varphi }^u$ under certain conditions.

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

Let $X$ be a Banach space, $ℬ\left(X\right)$ the algebra of bounded linear operators on $X$ and $(J,\parallel ·{\parallel }_J)$ an admissible Banach ideal of $ℬ\left(X\right)$. For $T\in ℬ\left(X\right)$, let $L_{J,T}$ and $R_{J,T}\in ℬ\left(J\right)$ denote the left and right multiplication defined by $L_{J,T}\left(A\right)=TA$ and $R_{J,T}\left(A\right)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in ℬ\left(X\right)$, and their elementary operators $L_{J,T}$ and $R_{J,T}$. In particular, we give necessary and sufficient conditions for $L_{J,T}$ and $R_{J,T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J,T}$ is recurrent...

Let $\varphi$ be a substitution of Pisot type on the alphabet $𝒜=\{1,2,...,d\}$; $\varphi$ satisfies theif for every $i,j\in 𝒜$, there are integers $k,n$ such that ${\varphi }^n\left(i\right)$ and ${\varphi }^n\left(j\right)$ have the same $k$-th letter, and the prefixes of length $k-1$ of ${\varphi }^n\left(i\right)$ and ${\varphi }^n\left(j\right)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d=2$ and provide a partial result for $d\ge 2$.

Let $(M,g)$ be a compact Riemannian manifold and $P_g$ an elliptic, formally self-adjoint, conformally covariant operator of order $m$ acting on smooth sections of a bundle over $M$. We prove that if $P_g$ has no rigid eigenspaces (see Definition 2.2), the set of functions $f\in C^{\infty }(M,ℝ)$ for which $P_{e^{f}g}$ has only simple non-zero eigenvalues is a residual set in $C^{\infty }(M,ℝ)$. As a consequence we prove that if $P_g$ has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics...

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...

The work is dedicated to investigating a limiting procedure for extending “local” integral operator equalities to “global” ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form $g\left(R_F\right)\int f_x\left(R_F\right)d\mu \left(x\right)=h\left(R_F\right)$. (1) They involve functions of the kind $X\ni x\mapsto f_x\left(R_F\right)\in B\left(F\right)$, where X is a general locally compact space, F runs over a suitable class of Banach subspaces...

Let $G$ be a simple algebraic group over an algebraically closed field $𝕜$ of characteristic 0, and $𝔤=LieG$. Let $(e,h,f)$ be an $𝔰{𝔩}_2$-triple in $𝔤$ with $e$ being a long root vector in $𝔤$. Let $(·,·)$ be the $G$-invariant bilinear form on $𝔤$ with $(e,f)=1$ and let $\chi \in {𝔤}^{*}$ be such that $\chi \left(x\right)=(e,x)$ for all $x\in 𝔤$. Let $𝒮$ be the Slodowy slice at $e$ through the adjoint orbit of $e$ and let $H$ be the enveloping algebra of $𝒮$; see [31]. In this article we give an explicit presentation of $H$ by generators and relations. As a consequence we deduce that $H$ contains...

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_i$ for which $\left(X_i\right)/ℐn\left(X_i\right)$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_i$ can be divided into subsets in such a way that if $X_i$ and $X_j$ are in different...

A cluster ensemble is a pair $(𝒳,𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $\Gamma$. The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p:𝒜\to 𝒳$. The space $𝒜$ is equipped with a closed $2$-form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.

In this paper we prove the following results. First, we show the existence of Wiener-Wintner dynamical system with continuous singular spectrum in the orthocomplement of their respective Kronecker factors. The second result states that if $f\in L^p$, $p$ large enough, is a Wiener-Wintner function then, for all $\gamma \in (1+\frac{1}{2p}-\frac{\beta }{2},1]$, there exists a set $X_f$ of full measure for which the series ${\sum }_{n=1}^{\infty }\frac{f\left(T^{n}x\right)e^{2\pi inϵ}}{n^{\gamma }}$ converges uniformly with respect to $ϵ$.

Let $X$ be a $\mu$-symmetric Hunt process on a LCCB space $𝙴$. For an open set $𝙶\subseteq 𝙴$, let ${\tau }_{𝙶}$ be the exit time of $X$ from $𝙶$ and $A^{𝙶}$ be the generator of the process killed when it leaves $𝙶$. Let $r:[0,\infty [\to [0,\infty [$ and $R\left(t\right)={\int }_0^{t}r\left(s\right)\mathrm{d}s$. We give necessary and sufficient conditions for ${𝔼}_{\mu }R\left({\tau }_{𝙶}\right)lt;\infty$ in terms of the behavior near the origin of the spectral measure of $-A^{𝙶}$. When $r\left(t\right)=t^l$, $l\ge 0$, by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l+1$ for ${\tau }_{𝙶}$...

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.