Some properties of m-isometries and m-invertible operators on banach spaces

We study left n-invertible operators introduced in two recent papers. We show how to construct a left n-inverse as a sum of a left inverse and a nilpotent operator. We provide refinements for results on products and tensor products of left n-invertible operators by Duggal and Müller (2013). Our study leads to improvements and different and often more direct proofs of results of Duggal and Müller (2013) and Sid Ahmed (2012). We make a conjecture about tensor products of left n-invertible operators and prove this conjecture in several cases. Finally, applications of these results are given to left n-invertible elementary operators and essentially left n-invertible operators.

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confidence: 58%

Section: Essential Left N-inversesmentioning

confidence: 99%

See 1 more Smart Citation

Structures of left n-invertible operators and their applications

2015

“…The study of m-left and m-right invertible operators has its roots in the work of Przeworska-Rolewicz [15,16], and has since been carried out by a number of authors, amongst them Sid Ahmed [17]. An interesting example of a left m-invertible Hilbert space operator is that of an m-isometric operator T for which m i=0 (−1) i ( m i )T * m−i T m−i = 0, where T * denotes the Hilbert space adjoint of T .…”

mentioning

confidence: 99%

Tensor product of left n-invertible operators

Duggal¹,

Müller

2013

A Banach space operator T ∈ B(X ) has a left m-inverse (resp., an essential left m-inverse) for some integer m ≥ 1 if there exists an operator S ∈ B(X ) (resp., an operator S ∈ B(X ) and a compact operator K ∈ B(X )) such thatis strictly left m-invertible (resp., strictly essentially left m-invertible), then T1 ⊗ T2 is: (i) left (m + n − 1)-invertible (resp., essentially left (m + n − 1)-invertible) if and only if T2 is left n-invertible (resp., essentially left n-invertible); (ii) strictly left (m + n − 1)invertible (resp., strictly essentially left (m + n − 1)-invertible) if and only if T2 is strictly left n-invertible (resp., strictly essentially left n-invertible).2010 Mathematics Subject Classification: Primary 47A80, 47A10; Secondary 47B47. Key words and phrases: Banach space, left n-invertible operator, essentially left ninvertible operator, tensor product, left-right multiplication operator.

“…As in Sid Ahmed [24] and Duggal and Müller [13], we say S is a left n-inverse of T (or T is a right n-inverse of S, or (S, T ) is an n-inverse pair ) if β n (S, T ) = 0. If β n (S, T ) = 0, but β n−1 (S, T ) = 0, we say S is a strict left n-inverse of T .…”

Section: Introductionmentioning

confidence: 99%

Tensor-splitting properties of $n$-inverse pairs of operators

Paul

2017

In this paper we study n-inverse pairs of operators on the tensor product of Banach spaces. In particular we show that an n-inverse pair of elementary tensors of operators on the tensor product of two Banach spaces can arise only from l-and m-inverse pairs of operators on the individual spaces. This gives a converse to a result of Duggal and Müller [13], and proves a conjecture of the second named author [16]. Our proof uses techniques from algebraic geometry, which generalize to other relations among operators in a tensor product. We apply this theory to obtain results for n-symmetries in a tensor product as well.