Resolvent algebra of finite rank operators
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Tusi Mathematical Research Group
ORIGINAL PAPER
Resolvent algebra of finite rank operators R. Eskandari1 • F. Mirzapour2 Received: 2 November 2019 / Accepted: 18 October 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract Let H be a Hilbert space. Suppose that A 2 BðHÞ and the operators I þ mA are invertible for all integers m 1. We characterize the resolvent algebra RA :¼ T 2 BðHÞ : sup kðI þ mAÞTðI þ mAÞ1 k\1 ; m1
when A is a finite rank operator with covðAÞ 6¼ 0. Moreover, we determine the elements of fAg0 and RcA0 and prove that RA ¼ RcA ¼ fAg0 RcA0 , where fAg0 is the commutant A and both RcA and RcA0 are subclasses of RA defined by n o RcA ¼ T 2 RA : lim kðI þ mAÞTðI þ mAÞ1 kexists m!1
and n o RcA0 ¼ T 2 RcA : lim kðI þ mAÞTðI þ mAÞ1 k ¼ 0 : m!1
We provide a counterexample showing that RcA ¼ fAg0 RcA0 is not true for some compact operators. Keywords Resolvent algebra Finite rank Commutant
Mathematics Subject Classification 47L30
Communicated by Evgenij Troitsky. & R. Eskandari [email protected] F. Mirzapour [email protected] 1
Department of Mathematics, Farhangian University, Tehran, Iran
2
Department of Mathematics, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran
R. Eskandari and F. Mirzapour
1 Introduction Let ðH; h; iÞ be a Hilbert space, and let BðHÞ denote the algebra of all bounded linear operators on H. A closed subspace M H is said to be invariant for a collection S BðHÞ if it is invariant for all A 2 S, i.e., AM M for each A 2 S. At the same token, M is called a hyperinvariant subspace of A if it is invariant for the comutant fAg0 :¼ fT 2 BðHÞ : TA ¼ AT g: The existence of hyperinvariant subspaces of a bounded linear operator on a Hilbert space is a long-standing problem, and has recently attracted the attention of a great number of mathematicians; see [9, 10, 12]. As a part of these investigations, Deddens [1] introduced the so-called Deddens algebra m m BA :¼ T 2 BðHÞ : sup kA TA k\1 m1
for an invertible linear operator A 2 BðHÞ. Clearly, fAg0 BA , and thus every invariant subspace of BA is also a hyperinvariant subspace of A. In [1, 16], the structure of BA is completely characterized. See also [4–6, 11, 13, 14] for more properties of BA . Suppose that the operators I þ mA are invertible for all integers m 1. The resolvent algebra is defined by 1 RA :¼ T 2 BðHÞ : sup kðI þ mAÞTðI þ mAÞ k\1 : m1
It is clear that fAg0 RA . In [7], it is shown that RA ¼ fAg0 when A is a nilpotent operator. An operator T 2 BðHÞ is called algebraic if there is a complex polynomial p such that pðTÞ ¼ 0. The polynomial of smallest degree annihilating T is called the characteristic polynomial of T. We consider the elements of BðHÞ with the special characteristic polynomial of the form pðxÞ ¼ xl ðx k1 Þ ðx kn Þ; where l 1; ki 6¼ 0 and fk1 ; . . .; kn g 6¼ ;. For a fixed l, we denote the class of all such elements by Bl ðHÞ. In [2], it was shown that RA possess nontrivial invariant subs
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