51 research outputs found
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,BâB(H), where B(H) is the algebra of operators on H into itself. Let δA,B: B(H)âB(H) denote the generalized derivation δAB(X)=AXâXB. 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, 1â¤p<â. We prove (i) If δAmBm(X)=0 for some XâB(H) and if either of A and B* is injective, then a necessary and sufficient condition for δAB(X)=0 is that ArXBnârâAnârXBr=0 for (any) two consecutive values of r,1â¤r<n. (ii) If δAmBm(X) and δAnBn(X)âCp for some XâB(H), and if m=2 or 3, then either δABn(X) or δABn+3(X)âCp; for general m and n, if A and B* are normal or subnormal, then there exists a natural number t such that δAB(X)âC2tnp. (iii) If δAmBm(X) and δAnBn(X)âCp for some XâB(H), and if either A is semi-Fredholm with ind Aâ¤0 or 1âA*AâCp, then δAB(X)âCp
Subspace gaps and range-kernel orthogonality of an elementary operator
AbstractRange-kernel orthogonality is established for certain elementary operators
Moving Weylâs theorem from â¨(T) to T
Schmoeger has shown that if Weyl's theorem holds for an isoloid Banach space operator T â B(X) with stable index, then it holds for â¨(T) whenever ⨠â Holo Ď (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 and The obtained solution is more general that
the obtained one for the classical solution since it is valid for an equation
of state while in the classical solution
Taking into account the current observations, we conclude
that must be a growing time function while 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 must be a growing time function if
is a positive decreasing function.Comment: 10 pages. RevTeX
Perturbations of operators satisfying a local growth condition
A Banach space operator Τ â () satisfies a local growth condition of order for some positive integer , Τ â loc(Gâ), if for every closed subset of the set of complex numbers and every x in the glocal spectral subspace Xâ () there exists an analytic function : Câ â such that (ΤâΝ¸) (Ν¸) ⥠and âĽ(Ν¸)âĽâ¤[dist(Îť; )]ÂŻÂŻÍŤ ÇÇ for some > 0 (independent of and ). Browder-Weyl type theorems are proved for perturbations by an algebraic operator of operators which are either loc(Gâ) or polynomially loc(Gâ).peerReviewe
Weyl's theorem and hypercyclic/supercyclic operators
AbstractNecessary and sufficient conditions for hypercyclic/supercyclic Banach space operators T to satisfy Ďa(T)âĎwa(T)=Ď00a(T) are proved
On the range closure of an elementary operator
AbstractLet B(H) denote the algebra of operators on a Hilbert H. Let ÎABâB(B(H)) and EâB(B(H)) denote the elementary operators ÎAB(X)=AXBâX and E(X)=AXBâCXD. We answer two questions posed by TurnĹĄek [Mh. Math. 132 (2001) 349â354] to prove that: (i) if A, B are contractions, then B(H)=ÎAB-1(0)âÎAB(B(H)) if and only if ÎABn(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)âE(B(H)) if and only if 0âisoĎ(E)
Functional equations and linear transformations - permutability and inversion. (Short Communication).
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