Converting the Properties in B(H) by Operators on B(H)

Abstract

The pair of operators on B(H) which are related to each other with respect to a specific property on B(H), have been studied before. In this paper, we study a pair of operators φ₁, φ₂ on B(H) which can convert some suitable properties to each other. For instance, we show that φ₁(T ) is a compact operator if and only if φ₂(T ) is compact, whenever φ₁(T ) is a Fredholm operator if and only if φ₂(T ) is a semi-Fredholm operator.

Key words and phrases. Preserving map, converting map, Fredholm operator, semi-Fredholm operator.

2010 Mathematics Subject Classification. Primary: 47B05; Secondary: 47L05, 47A53