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Abstract. We consider the linear maps $\varphi \colon \mathcal B(X)\to \mathcal B(Y)$ that preserve the semi-Fredholm operators in both directions or the essential spectrum of an operator, where $\mathcal B(X)$ is the algebra of all bounded linear operators on an infinite-dimensional Banach space $X$. We describe some known results in the Hilbert space case, provide some basic results and examples in the general case, and state several open problems.
DOI: 10.14232/actasm-017-327-x
AMS Subject Classification
(1991): 47B48, 47A10, 46H05
Keyword(s):
semi-Fredholm operators,
Calkin algebra,
linear preservers
Received November 30, 2017, and in revised form January 6, 2018. (Registered under 77/2017.)
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