The Fuglede-Putnam theory (Q2094620)

The research monograph under review is focused on the so-called Fuglede-Putnam-type theorems. The first result in this direction, dating back to 1942, is due to von Neumann. He proved that for any square matrices \(A\) and \(B\) of finite order, the difference of squares of the Frobenious norms of \([A, B]\) and \([A, B^*]\) is equal to the minus of the trace of \([A^*, A][B^*, B],\) where \([X, Y]\) denotes the cross-commutator of \(X\) and \(Y.\) An immediate consequence of this result shows that \([A, B]=0\) if and only if \([A, B^*]=0.\) The Fuglede-Putnam theorem, as stated below, generalizes this fact: Theorem. If \(T, A, B\) are a bounded linear operator on a Hilbert space \(H\) such that \(A\) and \(B\) are normal operators. Then \(TA=BT\) if and only if \(TA^*=B^*T.\) Indeed, there is a further generalization of this result when \(A\) and \(B\) are not necessarily bounded operators. The first chapter of this monograph collects variants of Fuglede-Putnam theorem including Berberian's trick. In the last section of Chapter 1, examples are presented explaining some obstructions in possible generalizations of the Fuglede-Putnam theorem. Chapter 2 deals with generalizations of the Fuglede-Putnam theorem when the operators \(A\) and \(B\) in question are not necessarily normal. Chapter 3 contains some asymptotic versions and Chapter 4 contains generalizations to Banach algebras. Chapter 5 is devoted to the case of unbounded nonnormal operators and Chapter 6 contains several applications of this theory. Chapter 7 is about further generalizations of the Fuglede-Putnam theorem and Chapter 8 contains several conjectures. This monograph has two appendices on bounded and unbounded operators. Overall, this monograph is informative, and it collects several interesting results from the single variable operator theory.