Generalized bicircular projections via the operator equation \(\alpha X^2AY^2+\beta XAY+A=0\) (Q947666)

A linear projection \(P\) on a complex Banach space \(E\) is called a generalized bicircular projection if, for some complex number \(\lambda \neq 1\) of modulus 1, the operator \(P+\lambda (I-P)\) is an isometry. Transformations of this kind have been rather extensively studied recently. Motivated by that study and the structure of surjective isometries on operator algebras, the author considers the following problem. Given a linear projection \(P\) on the space \(B(E)\) of all bounded linear operators and a complex number \(\lambda \neq 1\) of modulus 1, when does the mapping \(P+\lambda (I-P)\) equal one of the following transformations: \(A\mapsto UAV\), \(A\mapsto UA^*V\), \(A\mapsto UA^{t}V\)? Here, in the first case \(E\) is an arbitrary Banach space and \(U,V:E\to E\) are given bounded linear operators, in the second case \(E\) is a Hilbert space and \(U,V\) are conjugate-linear, while in the third case \(E\) is a Hilbert space and \(U,V\) are linear. Based on the solution of a certain operator equation, the structure of the transformation \(P\) is completely described in all those cases and some applications related to preserver problems are also given.