Hermitian indices and factorization of selfadjoint operators on a Kreĭn space
Newcastle University（纽卡斯尔大学）；Instituto Argentino de Matemática “Alberto P. Calderón” CONICET（阿尔贝托·P·卡尔德隆阿根廷数学研究所（CONICET））；University of Virginia（弗吉尼亚大学）
Michael A. Dritschel, Alejandra Maestripieri, James Rovnyak
The hermitian indices of a selfadjoint operator $C$ on a Kreĭn space $\mathcal H$ are defined as geometric measures of positivity and negativity of the operator. A different pair of indices arises in the Bognár-Krámli factorization of $C$, which writes $C$ as a product $AA^*$ where $A$ acts on a Kreĭn space $\mathcal A$ into $\mathcal H$ and has zero kernel; the new indices are the positive and negative indices of $\mathcal A$. Such factorizations are far from unique. When $\mathcal H$ is separable, it is known that the two notions of indices always coincide, and this has applications to index formulas in the theory of Julia operators and completion problems for operator matrices. A new proof of the equality of indices that does not require separability is given in this work.
Given a bounded selfadjoint operator W on a Krein space H and a closed subspace S of H, the Schur complement of W to S is defined under the hypothesis of weak complementability. A variational characterization of the Schur complement is given and the set of selfadjoint operators W admitting a Schur complement with these variational properties is shown to coincide with the set of S-weakly complementable selfadjoint operators.
We give an explicit and versatile parametrization of all positive selfadjoint extensions of a densely defined, closed, positive operator. In addition, we identify the Friedrichs extension by specifying the parameter to which it corresponds. This is a manuscript that was circulated as the first part of the preprint "Two papers on selfadjoint extensions of symmetric semibounded operators", INCREST Preprint Series, July 1981, Bucharest, Romania, but never published. In this LaTeX typeset version, only typos and a few inappropriate formulations have been corrected, with respect to the original man…
We consider densely defined sectorial operators $A_\pm$ that can be written in the form $A_\pm=\pm iS+V$ with $\mathcal{D}(A_\pm)=\mathcal{D}(S)=\mathcal{D}(V)$, where both $S$ and $V\geq \varepsilon>0$ are assumed to be symmetric. We develop an analog to the Birmin-Krein-Vishik-Grubb (BKVG) theory of selfadjoint extensions of a given strictly positive symmetric operator, where we will construct all maximally accretive extensions $A_D$ of $A_+$ with the property that $\overline{A_+}\subset A_D\subset A_-^*$. Here, $D$ is an auxiliary operator from $\ker(A_-^*)$ to $\ker(A_+^*)$ that parametriz…
Given B, C and W operators in the algebra L(H) of bounded linear operators on the Krein space H, the minimization problem min (BX - C)^#W(BX - C), for X in L(H), is studied when the weight W is selfadjoint. The analogous maximization and min-max problems are also considered. Complete answers to these problems and to those naturally associated to trace clase operators on Krein spaces are given.
The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space ${\mathfrak H}_1$ to a Hilbert space ${\mathfrak H}_2$ are considered in the Hilbert space ${\mathfrak H}_1\oplus{\mathfrak H}_2$ that contains the graph of $R$. They will be described by $2 \times 2$ blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation $S$ in ${\mathfrak H}_1 \oplus {\mathfrak H}_2$ that is induced by $R$. Such a relation is characterized by the orthogonality property ${\rm dom\,} S \perp {\rm ran\,} S$ and it is nonnegative. All nonnegative self…
This work deals with the functional model for a class of extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices. On the basis of these formulae, we are able to construct wave operators and derive a new representation for the scattering matrix for pairs of such extensions in both self-adjoint and non-self-adjoint situati…
Let ${\cal H}$ be a Hilbert space, $A$ a positive definite operator in ${\cal H}$ and $\langle f,g\rangle_A=\langle Af,g\rangle$, $f,g\in {\cal H}$, the $A$-inner product. This paper studies the geometry of the set $$ {\cal I}_A^a:=\{\hbox{ adjointable isometries for } \langle \ , \ \rangle_A\}. $$ It is proved that ${\cal I}_A^a$ is a submanifold of the Banach algebra of adjointable operators, and a homogeneous space of the group of invertible operators in ${\cal H}$, which are unitaries for the $A$-inner product. Smooth curves in ${\cal I}_A^a$ with given initial conditions, which are minima…
In this paper we initiate the study of a fundamental yet untapped random model of non-selfadjoint, bounded linear operators acting on a separable complex Hilbert space. We replace the weights $w_n=1$ in the classical unilateral shift $T$, defined as $Te_n=w_ne_{n+1}$, where $\{e_n\}_{n=1}^\infty$ form an orthonormal basis of a complex Hilbert space, by a sequence of i.i.d. random variables $\{X_n\}_{n=1}^{\infty}$; that is, $w_n=X_n$. This paper answers basic questions concerning such a model. We propose that this model can be studied in comparison with the classical Hardy/Bergman/Dirichlet sp…
Cones and gauges in complex spaces : Spectral gaps and complex Perron-Frobenius theory
我们引入复锥和相关的射影规范，推广了一个真正的伯克霍夫锥和希尔伯特度量复向量空间。在复巴拿赫空间中，我们推出了一系列谱间隙定理。证明了一个控制复锥压缩定理，并利用它将经典的Perron-Frobenius定理推广到复矩阵，将Jentzsch定理推广到复积分算子，将Krein-Rutman定理推广到紧和准紧的复算子，将Ruelle-Perron-Frobenius定理推广到动力系统的复转移算子.在一个复杂的n乘n矩阵A的最简单的情况下，我们有下面的陈述：假设0 < c < +无穷大使得|Im A_{ij} \bar{A}_{mn}| < c < Re A_{ij} \bar{A}_{mn}对于所有指数。那么A有一个“光谱间隙”。
