On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. I. General Case

Анотація

In the context of dissipative and accumulative differential equations (which contain the spectral parameter $\lambda$ nonlinearly) in a separable Hilbert space $\mathcal{H}$ we introduce a characteristic operator $M(\lambda)$ that works as an analog of the characteristic Weyl-Titchmarsh matrix. Its existence and properties are investigated. A description of $M(\lambda)$ that corresponds to separated boundary conditions is given. Analogs for Weyl functions and solutions are introduced. Weyl type inequalities for those analogs are established, which reduce to well-known inequalities in various special cases. The proofs are based on description and properties of maximal semi-definite subspaces in $\mathcal{H}^2$ of special form that we provide while studying boundary problems for equations as above.

Mathematics Subject Classification: 34A55, 34G10, 47E05.