Holomorphic Semigroups Research Articles

Tensor products of closed operators on Banach spaces

Let A and B be closed operators on Banach spaces X and Y. Assume that A and B have nonempty resolvent sets and that the spectra of A and B are unbounded. Let α be a uniform cross norm on X ⊗ Y. Using the Gelfand theory and resolvent algebra techniques, a spectral mapping theorem is proven for a certain class of rational functions of A and B. The class of admissable rational functions (including polynomials) depends on the spectra of A and B. The theory is applied to the cases A ⊗ I + I ⊗ B and A ⊗ B where A and B are the generators of bounded holomorphic semigroups.

A Characterization of Holomorphic Semigroups

A necessary and sufficient condition is given for a one-parameter semigroup $\{ U(t)\} ,\;0 \leqq t < \infty$, of class ${C_0}$ on a Banach space to be holomorphic (of class $H({\Phi _1},\;{\Phi _2})$ for some ${\Phi _1} < 0 < {\Phi _2}$). The condition is expressed in terms of the spectral properties of $U(t) - \zeta$ for small $t > 0$ and for a complex number $\zeta$ with $|\zeta | \geqq 1$.

A characterization of holomorphic semigroups

A necessary and sufficient condition is given for a one-parameter semigroup $\{ U(t)\} ,\;0 \leqq t < \infty$, of class ${C_0}$ on a Banach space to be holomorphic (of class $H({\Phi _1},\;{\Phi _2})$ for some ${\Phi _1} < 0 < {\Phi _2}$). The condition is expressed in terms of the spectral properties of $U(t) - \zeta$ for small $t > 0$ and for a complex number $\zeta$ with $|\zeta | \geqq 1$.

On numerical ranges and holomorphic semigroups

On numerical ranges and holomorphic semigroups

Properties of Quantum Statistical Expectation Values

The semigroup of statistical operators, and the unitary group of time translation operators generated by the same Hamiltonian of a nonrelativistic fermion field, are naturally imbedded in a holomorphic half-plane semigroup. The statistical expectation values of products of time-dependent operators are then boundary values of holomorphic functions which allow a Bochner integral representation with a Cauchy kernel. The general time-temperature-dependent Green's functions permit a concise spectral representation. It is suggested that a thermodynamic perturbation theory should treat the Heisenberg and Bloch equations simultaneously in terms of a perturbation theory for holomorphic half-plane semigroups generated by semibounded self-adjoint Hamiltonians.

Holomorphic semi-groups in a locally convex linear topological space

Holomorphic semi-groups in a locally convex linear topological space

A Summation Procedure for Certain Feynman Integrals

A mathematically rigorous summation procedure is developed for Feynman integrals involving a repulsive potential. The central idea of the proof is to connect certain Wiener integrals with the Feynman integral in question by making use of holomorphic vector-valued functions and holomorphic semigroups.