The Virozub-Matsaev condition and spectrum of definite type for self-adjoint operator functions
Let \(T:{\mathcal D}\to {\mathcal B}({\mathcal H})\) be a self-adjoint operator function, that is, \(T(\lambda^*)=T(\lambda)^*,\;\lambda\in {\mathcal D}\), where \({\mathcal D}\) is some subset of the complex plane, symmetric with respect to \(\mathbb{R}\) and \({\mathcal B}({\mathcal H})\) is the space of bounded linear operators in a Hilbert space \({\mathcal H}\). For the continuously differentiable self-adjoint operator function \(T\), the Virozub-Matsaev condition (VM) is said to hold in a point \(\lambda_0\in {\mathcal D}\cap \mathbb{R}\) [on an interval \(\Delta\subset {\mathcal D}\cap \mathbb{R}\), resp.] if there exist positive constants \(\varepsilon\) and \(\delta\) such that \[ \|x\|=1,\;|(T(\lambda_0)x,x)|\leq\varepsilon\Longrightarrow (T'(\lambda_0)x,x)\geq \delta\tag{*} \] [if (*) holds for each \(\lambda_0\in\Delta\), resp.]. The condition (VM) is closely connected with the notion of a spectral point of positive type of the operator function \(T\) and to the existence and some properties of a local spectral function \(Q_t\) of \(T\) on an interval \(\Delta\) subject to \(\int_\Delta T(t)\,dQ_t=0\). Here, \(\lambda_0\in\sigma(T)\) is of positive type if (*) is in force with \(\|T(\lambda_0)x\|\) in place of \(|(T(\lambda_0)x,x)|\) therein. Following the notion and some results of the first and third authors and \textit{V.\,Matsaev} [J.~Funct.\ Anal.\ 235, No.\,1, 193--225 (2006; Zbl 1125.47010)], the authors of the present paper establish sufficient conditions for the condition (VM) to hold for twice continuously differentiable \(T\). Among others, the numerical range \(W(T)\) of \(T\) and its real subset \(W(T)\cap \mathbb{R}\) in conjunction with the so-called condition (vm), which is weaker and much simpler than (VM), play a key role. The following theorem is one of the main results. Let \(T\) be twice continuously differentiable on an open interval \(I=(\mu,\nu)\subset \mathbb{R}\) with respect to the operator norm, and let \(\omega(I)\in \bar{I}\) be defined by \[ \omega(I)=\begin{cases} \sup W(T) &\quad \text{if \(W(T)\neq \varnothing\)},\\ \nu &\quad \text{if \((T(\lambda)x,x)<0\) for all \(\lambda\in I,\;0\neq x\in {\mathcal H}\)},\\ \mu &\quad \text{if \((T(\lambda(\lambda)x,x))>0\) for all \(\lambda \in I,\;0\neq x\in {\mathcal H}\)}. \end{cases} \] Assume that the condition (VM) is in force on \(I\). If \(\lambda_0\in I\), \(\lambda_0\leq \omega(I)\), and there exists a constant \(c_{\lambda_0}>0\) such that \[ \max\{(T'(\lambda_0)x,x),(T''(\lambda_0)x,x)\}\geq c_{\lambda_0}\;\text{\;for each \(x\in {\mathcal H}\), \(\|x\|=1\)}, \] then \(T\) satisfies the condition (VM) in \(\lambda_0\). Applications to self-adjoint operators in Krein spaces and to quadratic operator polynomials are given as well.
scientific article; zbMATH DE number 5320870