Trace formulae and principal functions of Hilbert space operators(Recent Developments in Linear Operator Theory and its Applications)
Muneo Chō · Institutional Repositories DataBase (IRDB) · 2005
This paper is the results of [13], [14] and [15].Let $\mathcal{H}$ be a complex separable Hilbert space and $B$ (-?) be the set of all bounded linear operators on $\mathcal{H}$ .About the $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$ formula, we have the following:Theorem 1 (M.Krein, 1953).Let $A$ be a self-adjoint operator on $\mathcal{H}$ and $K$ be $a$ trace class self-adjoint operator on $\mathcal{H}$ .Then there exists a unique function $\delta(t)$ such that Tr $(p(A+K)-p(A))= \int p'(t)\delta(t)dt$ ,where $p$ is a polynomial.Let $\mathrm{C}_{1}$ be the $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$ class and $A$ be the set of all Laurent polynomials; $P(r, z)=$ $\sum_{k=-N}^{N}p_{k}(r)z^{k}$ .Let $\mathrm{J}(\mathrm{p}, Q)$ be the Jacobian of $P$ , $Q$ .Theorem 2 , Helton-Howe [19]).Let $T=X+iY$ be an operator on $\mathcal{H}$ with trace class self-commutator $([T^{*}, T]\in \mathrm{C}_{1})$ .Then there eists a functiony)dxdy$ , where $p$ and $q$ are polynomials of two variables.Functions $\delta(t)$ and $g(x, y)$ irx Theorems 1 and 2 axe called the phase shift of the perturbation problem $Aarrow A+K$, and the (Cartesian) principal function of $T$ , respectively.Let $T$ be hyponormal and satisfy $[T^{*},T]\in \mathrm{C}_{1}$ .For operators $A$ and $K$ of Theorem 1, let $A=TT^{*}$ and $K=T^{*}T-TT^{*}(=[T^{*},T]\in \mathrm{C}_{1})$ .Then Theorem 1 is $\mathrm{T}\mathrm{r}(p(T^{*}T)-p(TT^{*}))=\int p'(t)\delta(t)dt$ .