Kato's equality and spectral decompositon for positive \(C_ 0\)-groups
Let E be a Banach lattice, and let \(\{T(t):t\geq 0\}\) be a semigroup of class \(C_ 0\) on E with the generator A. \textit{R. Nagel} and \textit{H. Uhlig} proved that if E has order continuous norm, then T(t) is a lattice homomorphism for all \(t\geq 0\) if and only if the (generalized) Kato equality \(A| x| =sign(x)Ax\quad(x\in D(A))\) holds, where sign(x) denotes the difference of the band projections onto \((x^+)^{\perp \perp}\), resp. \((x^-)^{\perp \perp}\) [cf. J. Oper. Theory 6, 113-123 (1981; Zbl 0486.47025)]. The main result of the paper under review is that, for an arbitrary E, T(t) is a lattice homomorphism for all \(t\geq 0\) if and only if the following weak form of Kato's equality \[ <| x|,A'x'>=<sign(q(x))Ax,x'>\quad(x\in D(A),\quad x'D(A')) \] holds, where \(q:E\to E''\) denotes the evaluation map. Kato's equality is then used to obtain: Let A be the generator of a positive \(C_ 0\)-group on a Banach lattice E. If m is a real number in the resolvent of A, then E is the direct sum of two orthogonal bands \(I_ m\) and \(J_ m\) invariant under the group and such that the restrictions of A to them have spectra contained in \(\sigma\) (A) and in Re \(z\lessgtr m\), respectively. In the last section semigroups of lattice homomorphisms in C(X) and their spectral properties are studied (where X is compact).
scientific article; zbMATH DE number 3863921
scientific article