The Commutant Modulo Cp of Co-prime Powers of Operators on a Hilbert Space

AbstractLet H be a separable infinite-dimensional complex Hilbert space and let A,B∈B(H), where B(H) is the algebra of operators on H into itself. Let δA,B: B(H)→B(H) denote the generalized derivation δAB(X)=AX−XB. This note considers the relationship between the commutant of an operator and the commutant of co-prime powers of the operator. Let m,n be some co-prime natural numbers and let Cp denote the Schatten p-class, 1≤p<∞. We prove (i) If δAmBm(X)=0 for some X∈B(H) and if either of A and B* is injective, then a necessary and sufficient condition for δAB(X)=0 is that ArXBn−r−An−rXBr=0 for (any) two consecutive values of r,1≤r<n. (ii) If δAmBm(X) and δAnBn(X)∈Cp for some X∈B(H), and if m=2 or 3, then either δABn(X) or δABn+3(X)∈Cp; for general m and n, if A and B* are normal or subnormal, then there exists a natural number t such that δAB(X)∈C2tnp. (iii) If δAmBm(X) and δAnBn(X)∈Cp for some X∈B(H), and if either A is semi-Fredholm with ind A≤0 or 1−A*A∈Cp, then δAB(X)∈Cp

Subspace gaps and range-kernel orthogonality of an elementary operator

AbstractRange-kernel orthogonality is established for certain elementary operators

Moving Weyl’s theorem from ⨍(T) to T

Schmoeger has shown that if Weyl's theorem holds for an isoloid Banach space operator T ∈ B(X) with stable index, then it holds for ⨍(T) whenever ⨍ ∈ Holo σ (T) is a function holomorphic on some neighbourhood of the spectrum of T. In this note we establish a converse.peerReviewe

Bianchi II with time varying constants. Self-similar approach

We study a perfect fluid Bianchi II models with time varying constants under the self-similarity approach. In the first of the studied model, we consider that only vary $G$ and $\Lambda.$ The obtained solution is more general that the obtained one for the classical solution since it is valid for an equation of state $\omega\in(-1,\infty)$ while in the classical solution $\omega\in(-1/3,1) .$ Taking into account the current observations, we conclude that $G$ must be a growing time function while $\Lambda$ is a positive decreasing function. In the second of the studied models we consider a variable speed of light (VSL). We obtain a similar solution as in the first model arriving to the conclusions that $c$ must be a growing time function if $\Lambda$ is a positive decreasing function.Comment: 10 pages. RevTeX

Perturbations of operators satisfying a local growth condition

A Banach space operator Τ ∊ () satisfies a local growth condition of order for some positive integer , Τ ∊ loc(Gₘ), if for every closed subset of the set of complex numbers and every x in the glocal spectral subspace Xₜ () there exists an analytic function : C⟍ → such that (Τ―λ¸) (λ¸) ≡ and ∥(λ¸)∥≤[dist(λ; )]¯¯ͫ ǁǁ for some > 0 (independent of and ). Browder-Weyl type theorems are proved for perturbations by an algebraic operator of operators which are either loc(Gₘ) or polynomially loc(Gₘ).peerReviewe

Weyl's theorem and hypercyclic/supercyclic operators

AbstractNecessary and sufficient conditions for hypercyclic/supercyclic Banach space operators T to satisfy σa(T)∖σwa(T)=π00a(T) are proved

On the range closure of an elementary operator

AbstractLet B(H) denote the algebra of operators on a Hilbert H. Let ΔAB∈B(B(H)) and E∈B(B(H)) denote the elementary operators ΔAB(X)=AXB−X and E(X)=AXB−CXD. We answer two questions posed by Turnšek [Mh. Math. 132 (2001) 349–354] to prove that: (i) if A, B are contractions, then B(H)=ΔAB-1(0)⊕ΔAB(B(H)) if and only if ΔABn(B(H)) is closed for some integer n⩾1; (ii) if A, B, C and D are normal operators such that A commutes with C and B commutes with D, then B(H)=E-1(0)⊕E(B(H)) if and only if 0∈isoσ(E)