In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂={𝕂}^{+}\oplus {𝕂}^{-}$ of the Krein space $𝕂$ with ${𝕂}^{+}$ and ${𝕂}^{-}$ invariant under $T$.

We prove that for the spectral radius of a weighted composition operator $a{T}_{\alpha }$, acting in the space $L^p(X,,\mu )$, the following variational principle holds: $lnr\left(a{T}_{\alpha }\right)=ma{x}_{\nu \in M{¹}_{\alpha ,e}}{\int }_{X}ln\left|a\right|d\nu$, where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, $M{¹}_{\alpha ,e}$ is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and ${}_{\infty }$-measurable function, where ${}_{\infty }={\bigcap }_{n=0}^{\infty }{\alpha }^{-n}\left(\right)$. This considerably extends the range of validity of the above formula, which was previously known...

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.

We introduce the definition of $p$-limited completely continuous operators, $1\le p<\infty$. The question of whether a space of operators has the property that every $p$-limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$-limited completely continuous evaluation operators.

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.

Let X be a closed subspace of $L^p\left(\mu \right)$, where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $su{p}_{n\in \mathbb{Z}}\left|\right|Uⁿ\left|\right|<\infty$. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like ${\sum }_{n\ge 1}\left(Uⁿf\right)/n^{1-\alpha }$, 0 ≤ α < 1, in terms of $\left|\right|f+\cdots +U^{n-1}f{\left|\right|}_p$, generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. ...

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...

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...

Let α be an isometric automorphism of the algebra ${}_p$ of bounded linear operators in ${}^p[0,1]$ (p ≥ 1). Then α transforms conditional expectations into conditional expectations if and only if α is induced by a measure preserving isomorphism of [0, 1].

Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w*}(X*,Y)$, as well as the complementation of the space $K_{w*}(X*,Y)$ of w*-w compact operators in the space $L_{w*}(X*,Y)$ of w*-w operators from X* to Y.

We investigate Banach space automorphisms $T:{\ell }_{\infty }/c₀\to {\ell }_{\infty }/c₀$ focusing on the possibility of representing their fragments of the form $T_{B,A}:{\ell }_{\infty }\left(A\right)/c₀\left(A\right)\to {\ell }_{\infty }\left(B\right)/c₀\left(B\right)$ for A,B ⊆ ℕ infinite by means of linear operators from ${\ell }_{\infty }\left(A\right)$ into ${\ell }_{\infty }\left(B\right)$, infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on ${\ell }_{\infty }/c₀$. We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for...

Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^p\left(\nu \right)$, $T₁,...,T_k$, $n̅=(n₁,...,n_k)\in {\mathbb{N}}^k$ and $\alpha ̅=(\alpha ₁,...,{\alpha }_k)$ with $0<{\alpha }_j\le 1$, we define the ergodic Cesàro-α̅ averages ${}_{n̅,\alpha ̅}f=1/\left({\prod }_{j=1}^k{A}_{n_j}^{{\alpha }_j}\right){\sum }_{i_k=0}^{n_k}\cdots {\sum }_{i₁=0}^{n₁}{\prod }_{j=1}^k{A}_{n_j-i_j}^{{\alpha }_j-1}{T}_k^{i_k}\cdots T{₁}^{i₁}f$. For these averages we prove the almost everywhere convergence on X and the convergence in the $L^p\left(\nu \right)$ norm, when $n₁,...,n_k\to \infty$ independently, for all $f\in L^p\left(d\nu \right)$ with p > 1/α⁎ where $\alpha ⁎=mi{n}_{1\le j\le k}{\alpha }_j$. In the limit case p = 1/α⁎, we prove that the averages ${}_{n̅,\alpha ̅}f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $\Lambda (1/\alpha ⁎,{\phi }_{m-1})$ with $\phi ₘ\left(t\right)=t{(1+log⁺t)}^m$. To obtain the result in the limit case we need...

Let $U₁,...,U_d$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $\mathbb{Z}{₊}^d$ and the associated collection of rectangular parallelepipeds in ${ℝ}^d$ with sides parallel to the axes and dimensions of the form $n₁\times \cdots \times n_d$ with $(n₁,...,n_d)\in \Gamma .$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{\Gamma }$ are defined respectively on $L¹\left({ℝ}^d\right)$ and L¹(Ω) by $M_{}g\left(x\right)=su{p}_{x\in R\in }1/\left|R\right|{\int }_R\left|g\left(y\right)\right|dy$ and $M_{\Gamma }f\left(\omega \right)=su{p}_{(n₁,...,n_d)\in \Gamma }1/n₁\cdots n_d{\sum }_{j₁=0}^{n₁-1}\cdots {\sum }_{j_d=0}^{n_d-1}\left|f\left(U{₁}^{j₁}\cdots U_d^{j_d}\omega \right)\right|$. Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate ...

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

Let $u$ be a holomorphic function and $\varphi$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u{C}_{\varphi }$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$, $\varphi$, their derivatives, and ${\varphi }^n$, the $n$-th power of $\varphi$. Moreover, we obtain some similar estimates for the essential norms of the operators $u{C}_{\varphi }$, from which sufficient and necessary conditions of compactness of $u{C}_{\varphi }$ follows immediately. ...

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 $\left(\mathrm{SBaw}\right)$,...

We investigate the $L_p$-spectrum of linear operators defined consistently on $L_p\left(\Omega \right)$ for p₀ ≤ p ≤ p₁, where (Ω,μ) is an arbitrary σ-finite measure space and 1 ≤ p₀ < p₁ ≤ ∞. We prove p-independence of the $L_p$-spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω,μ); the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition. We show how previous results on $L_p$-spectral independence...