Smooth maximally dissipative boundary-value problems for a parabolic equation in a Hilbert space
Let H be a separable Hilbert space, A:\(H\to H\) a linear, selfadjoint, positive operator. The minimal and maximal extensions \(L_ 0\) and L of the differential operator \(\ell(y)=y'(t)+Ay(t)\) in the space \(L_ 2(H[0,b])\) are defined. The main result concerns the nature of the elements of the domain \(D(L_ 0)\) and D(L). Let \(D_ H(L)\subset D(L)\) be the set of all vector functions which take some well defined boundary values at \(t=0\) and \(t=b\), and let \(L_ H\) be the restriction of the operator L to the set \(D_ H(L)\). Define \(\tilde L\) to be a maximal dissipative extension of \(L_ 0\) such that \(L_ 0\subset \tilde L\subset L_ H\). It is proven that any operator of this kind is an extension \(L_ K\) of certain differential operator \({\tilde \ell}(y)=y'(t)+\hat Ay(t),\) subject to the boundary condition of the form \(y(0)=Ky(b),\) where K is some generalized contraction in H \((\| K\| \leq 1)\), and that any operator \(L_ K\) is a maximal dissipative extension of \(L_ 0\), such that \(L_ 0\subset L_ K\subset L_ H\). Some spectral properties of operators \(L_ K\) are discussed. Namely it is proven that if \((A+\mu E)^{-{1\over2}}\), for some fixed \(\mu>0\), is an operator of Hilbert-Schmidt class, then the resolvent of \(L_ K\) is of the Hilbert-Schmidt class too.
scientific article; zbMATH DE number 3859556