Generalized dissipativeness in a Banach space (Q1907696)

Summary: Suppose \(X\) is a real or complex Banach space with dual \(X^*\) and a semiscalar product \([ , ]\). For \(k\) a real number, a subset \(B\) of \(X\times X\) will be called \(k\)-dissipative if for each pair of elements \((x_1, y_1)\), \((x_2, y_2)\) in \(B\), there exists \(h\in \{f\in X^*: [x, f]= |x|^2= |f|^2\}\) such that \(\text{Re}[y_1- y_2, h]\leq k|x_1- x_2|^2\). This definition extends a notion of dissipativeness which is equivalent to having \(k\) equal zero here. A number of definitions and theorems related to this original dissipative notion are generalized in the present paper to fit the \(k\)-dissipative situation, and proofs are given for the new theorems.