Products of roots of the identity

Journal of Mathematical Analysis and Applications, 1971

arXiv (Cornell University), 2018

The primary purpose of this paper is to show the existence of normal square and nth roots of some classes of bounded operators on Hilbert spaces. Two interesting simple results hold. Namely, under simple conditions we show that if any operator T is such that T 2 = 0, then this implies that T is normal and so T = 0. Also, we will see when the square root of an arbitrary bounded operator is normal.

We give a necessary and sufficient condition for a certain set of infinite products of linear operators to be zero. We shall investigate also the case when this set of infinite products converges to a non-zero operator.

arXiv (Cornell University), 2020

We show that a densely defined closable operator A such that the resolvent set of A 2 is not empty, is necessarily closed. This result is then extended to the case of a polynomial p(A). We also generalize a recent result by Sebestyén-Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given, one of them being a proof that if T is a quasinormal (unbounded) operator such that T n is normal for some n ≥ 2, then T is normal. By a recent result by Pietrzycki-Stochel, we infer that a closed subnormal operator such that T n is normal, must be normal. Another remarkable result is the fact that a hyponormal operator A, bounded or not, such that A p and A q are self-adjoint for some co-prime numbers p and q, is self-adjoint. It is also shown that an invertible operator (bounded or not) A for which A p and A q are normal for some co-prime numbers p and q, is normal. These two results are shown using Bézout's theorem in arithmetic. Notation First, we assume that readers have some familiarity with the standard notions and results in operator theory (see e.g. [17] and [25] for some background). We do recall most of the needed notions though. First, note that in this paper all operators are linear. Let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear operators defined from H into H. If S and T are two linear operators with domains D(S) ⊂ H and D(T) ⊂ H respectively, then T is said to be an extension of S, written S ⊂ T , when D(S) ⊂ D(T) and S and T coincide on D(S). The product ST and the sum S + T of two operators S and T are defined in the usual fashion on the natural domains: D(ST) = {x ∈ D(T) : T x ∈ D(S)} and D(S + T) = D(S) ∩ D(T).

Journal of Mathematical Analysis and Applications, 2013

Inspired by the well-known Grüss inequality, we study the multiplicativity of linear operators, satisfying certain conditions. Applications to some classical operators will be given.

Journal of Mathematical Analysis and Applications, 1990

Analysis Mathematica, 2020

The primary purpose of this paper is to investigate the question of invertibility of the sum of operators. The setting is bounded and unbounded linear operators. Some interesting examples and consequences are given. As an illustrative point, we characterize invertibility for the class of normal operators. Also, we give a very short proof of the self-adjointness of a normal operator when the latter has a real spectrum.

Linear Algebra and its Applications, 2006

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce the class, denoted QA, of operators satisfying T * |T 2 |T T * |T | 2 T and we prove basic structural properties of these operators. Using these results, we also prove that if E is the Riesz idempotent for a non-zero isolated point λ 0 of the spectrum of T ∈ QA, then E is self-adjoint, and we give a necessary and sufficient condition for T ⊗ S to be in QA when T and S are both non-zero operators.

Pacific Journal of Mathematics, 1976

commutes with * that is, * = * and it is denuded by [nQN]. In this paper we investigate some properties of n-power quasinormal operators. Also, the necessary and sufficient condition for a Binormal operator to be 2 power quasi normal operator is obtained. Mathematics Subject Classification: 47B20 Keywords: Self adjoint operator, n -power quasi normal operator, unitary and binormal operator.