A. A. Shkalikov
We relax assumptions for a dissipative operator in Krein space to possess a maximal non-negative invariant subspace. Our main result is a generalization of a well-known Pontrjagin-Krein-Langer-Azizov theorem. Then we investigate the semigroup properties of the restriction of the operator onto the invariant subspace.
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We study unitary representations of groups in Krein spaces, irreducibility criteria and integral decompositions. Our main tool is the theory of Krein subspaces and their (reproducing) kernels and a variant of Choquet's theorem.
In this paper we provide a study of quaternionic inner product spaces. This includes ortho-complemented subspaces, fundamental decompositions as well as a number of results of topological nature. Our main purpose is to show that a uniformly positive subspace in a quaternionic Krein space is ortho-complemented, and this leads to our choice of the results presented in the paper.
Rearrangement Invariant Orthogonal Sums in Krein Spaces. II
本文研究了Krein空间中伪正则子空间的无穷正交和定义及其条件，通过无条件和Moore-Smith和进行分析。
Extensions of dual definite subspaces to dual maximal definite ones are described. The concepts of dual quasi maximal subspaces and quasi basis are introduced and studied. The obtained results are applied to the classification of C-symmetries.
We discuss Lomonosov's proof of the Pontryagin-Krein Theorem on invariant maximal non-positive subspaces, prove the refinement of one theorem from \cite{OShT} on common fixed points for a group of fractional-linear maps of operator ball and deduce its consequences. Some Burnside-type counterparts of the Pontryagin-Krein Theorem are also considered.
The infinitesimal generator $A$ of a strongly continuous semigroup on a Hilbert space is assumed to satisfy that $B_β:=A-β$ is a sectorial operator of angle less than $\fracπ{2}$ for some $β\geq 0$. If $B_β$ is dissipative in some equivalent scalar product then the Naimark-Arocena Representation Theorem is applied to obtain a Kre\uın space unitary dilation of the semigroup.
A $J$-frame for a Krein space $\mathcal{H}$ is in particular a frame for $\mathcal{H}$ (in the Hilbert space sense). But it is also compatible with the indefinite inner-product of $\mathcal{H}$, meaning that it determines a pair of maximal uniformly definite subspaces, an analogue to the maximal dual pair associated to an orthonormal basis in a Krein space. This work is devoted to study duality for $J$-frames in Krein spaces. Also, tight and Parseval $J$-frames are defined and characterized.
M. Krein proved in 1948 that if T is a continuous operator on a normed space leaving invariant an open cone, then its adjoint T* has an eigenvector. We present generalizations of this result as well as some applications to C*-algebras, operators on l_1, operators with invariant sets, contractions on Banach lattices, the Invariant Subspace Problem, and von Neumann algebras.
In this paper we characterize $\sqrt{2}$-1-uniform $J$-Parseval fusion frames in a Krein space $\mathbb{K}$. We provide a few results regarding construction of new $J$-tight fusion frame from given $J$-tight fusion frames. We also characterize any uniformly $J$-definite subspace of a Krein space $\mathbb{K}$ in terms of a $J$-fusion frame inequality. Finally we generalize the fundamental identity of frames in Krein space $J$-fusion frame setting.
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.
Definition. Let J be a period-2 unitary operator (some people say J is reflection operator or reflection symmetry) and U be a linear operator. If U^*JU = J (resp. U^*JU >= J) then U is said to be J-isometry (resp. J-noncontraction). If U^*JU >= J and UJU^* >= J) then U is said to be J-binoncontraction). Theorem. If every J-isometry has nontrivial positive invariant subspace then every J-noncontraction has such a subspace. Theorem. If every J-binoncontractive J-isometry has maximal positive invariant subspace then every J-noncontraction has such a subspace. The article text is the complete text…
It is shown that if $X$ is a unitary operator so that a singular subspace of~$U$ is unitarily equivalent to a singular subspace of~$UX$ (or $XU$), for each unitary operator~$U$, then $X$ is the identity operator. In other words, there is no nontrivial generalization of Birman-Krein Theorem that includes the preservation of a singular spectral subspace in this context.
A definition of frames for Krein spaces is proposed, which extends the notion of $J$-orthonormal basis of Krein spaces. A $J$-frame for a Krein space $(\HH, \K{\,}{\,})$ is in particular a frame for $\HH$ in the Hilbert space sense. But it is also compatible with the indefinite inner product $\K{\,}{\,}$, meaning that it determines a pair of maximal uniformly $J$-definite subspaces with different positivity, an analogue to the maximal dual pair associated to a $J$-orthonormal basis. Also, each $J$-frame induces an indefinite reconstruction formula for the vectors in $\HH$, which resembles the …
Let $\{f_n:n\in\mathbb{N}\}$ be a $J$-frame for a Krein space ${\textbf{\textit{K}}}$ and $P_M$ be a $J$-orthogonal projection from ${\textbf{\textit{K}}}$ onto a subspace $M$. In this article we find sufficient conditions under which $\{P_M(f_n):n\in\mathbb{N}\}$ is a $J$-frame for $P_M\textbf{\textit{K}}$ and $\{(I-P_M)f_n\}_{n\in{\mathbb{N}}}$ is a $J$-frame for $(I-P_M)\textbf{\textit{K}}$. We also introduce $J$-frame sequence for a Krein space ${\textbf{\textit{K}}}$ and study some properties of $J$-frame sequence analogues to Hilbert space frame theory.
In this article we find a necessary and sufficient condition under which a given collection of subspace is a $J$-fusion frame for a Krein space $\mathbb{K}$. We also approximate $J$-fusion frame bounds of a $J$-fusion frame by the upper and lower bounds of the synthesis operator. Then, we obtain the $J$-fusion frame bounds of the cannonical $J$-dual fusion frame. Finally, we address the problem of characterizing those bounded linear operators in $\mathbb{K}$ for which the image of $J$-fusion frame is also a $J$-fusion frame.
We describe necessary and sufficient conditions for a $J$-dissipative operator in a Krein space to have maximal semidefinite invariant subspaces. The semigroup properties of the restrictions of an operator to these subspaces are studied. Applications are given to the case when an operator admits matrix representation with respect to the canonical decomposition of the space. The main conditions are given in the terms of the interpolation theory of Banach spaces.
We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given.
The theorem on the existence of maximal nonnegative invariant subspaces for a special class of dissipative operators in Hilbert space with indefinite inner product is proved in the paper. It is shown in addition that the spectra of the restrictions of these operators on the corresponding invariant subspaces lie in the closed upper half-plane. The obtained theorem is a generalization of well-known results of L.S.Pontrjagin, H.K.Langer, M.G.Krein and T.Ja.Azizov devoted to this subject.
Theorem 1. Given a number c >= 1, there exists a J-unitary operator \hat{V}, such that: (a) r(\hat{V})= r(\hat{V}^{-1})= c ; (b) S(c^{-1}\hat{V})=S(c^{-1}\hat{V}^{-1}) =S(c^{-1}\hat{V}^{*-1}) = S(c^{-1}\hat{V}^*)={0} (c) there exist maximal strictly positive and strictly negative \hat{V}^{\pm 1}-{invariant subspaces} L_{+}, L_{-}, such that they are mutually J-orthogonal and L_{+} + L_{-} is dense in the space. (d_1) if L_1 is non-zero \hat{V}-invariant subspace, then r(\hat{V}|L_1)=r(\hat{V}) (d_2) if L_2 is non-zero \hat{V}^{-1}-invariant subspace, then r(\hat{V}^{-1}|L_2)=r(\hat{V}^{-1}) .
If $\left(\h,\langle\cdot,\cdot\rangle\right)$ is a Hilbert space and on it we consider the sesquilinear form $\langle\,W\cdot,\cdot\rangle$ so-called $W$-metric, where $W^{*}=W\in\BH$, and $\ker\,W=\{0\}$, then the space $\left(\h,\langle\,W\cdot,\cdot\rangle\right)$ is called Hilbert space with $W$-metric or simply $W$-space. In this paper we investigate the dynamic of frames of subspace on these spaces, where the sense of dynamics refers to the behavior of frames of subspace in $\h_{W}$ (the completion of $\left(\h,\langle\,W\cdot,\cdot\rangle\right)$) comparing with $\h$ and vice versa. Th…
Let $\mathcal{H}$ be a Krein space with fundamental symmetry $J$. Along this paper, the geometric structure of the set of $J$-normal projections $\mathcal{Q}$ is studied. The group of $J$-unitary operators $\mathcal{U}_J$ naturally acts on $\mathcal{Q}$. Each orbit of this action turns out to be an analytic homogeneous space of $\mathcal{U}_J$, and a connected component of $\mathcal{Q}$. The relationship between $\mathcal{Q}$ and the set $\mathcal{E}$ of $J$-selfadjoint projections is analized: both sets are analytic submanifolds of $L(\mathcal{H})$ and there is a natural real analytic submers…
A definition of frames in Krein spaces is proposed which extends the concept of $J$-frames defined by J.I. Giribet et al., J. Math. Anal. Appl. ${\textbf{393}}$ (2012), 122-137. The principal difference consists in the fact that a $J$-frame is related to maximal definite subspaces $\mathcal{M}_\pm$ which are not assumed to be uniformly definite. The latter allows one to extend the collection of $J$-frames. In particular, some complete $J$-orthogonal sequences and $J$-orthogonal Schauder bases can be interpreted as $J$-frames.
Locus of non-real eigenvalues of a class of linear relations in a Krein space
本文研究Krein空间中一类线性关系的非实特征值位置，将经典极大对称算子特征值虚部上界中的常数从2改进为约1.84，并推广到闭对称关系及其耗散扩张。
G-strong subdifferentiability and applications to norm attaining subspaces
本文在群不变映射框架下研究自反性与强次可微性，证明对偶范数G-强次可微蕴含Banach空间G-自反，并推广多个经典泛函分析定理，最终给出范数可达泛函集合中存在闭Banach子空间的充分条件。
Trace formulas in higher dimensions
本文推导了多变量算子函数在对称范数理想上的Krein和Koplienko迹公式，涵盖自伴及最大耗散算子，适用于正常和奇异迹的理想，并通过实例展示结果。
For a bounded Hankel matrix $Γ$, we describe the structure of the Schmidt subspaces of $Γ$, namely the eigenspaces of $Γ^* Γ$ corresponding to non zero eigenvalues. We prove that these subspaces are in correspondence with weighted model spaces in the Hardy space on the unit circle. Here we use the term "weighted model space" to describe the range of an isometric multiplier acting on a model space. Further, we obtain similar results for Hankel operators acting in the Hardy space on the real line. Finally, we give a streamlined proof of the Adamyan-Arov-Krein theorem using the language of weight…
It is known that (i) a subspace ${\mathcal N}$ of the Hardy space $H^2$ which is invariant under the backward shift operator can be represented as the range of the observability operator of a conservative discrete-time linear system, (ii) the transfer-function of this conservative linear system in turn is the inner Beurling-Lax representer for the forward-shift invariant subspace ${\mathcal M} : = {\mathcal N}^\perp$, and (iii) this transfer function also serves as the Sz.-Nagy-Foias characteristic function of the pure contraction operator $T$ given by $T = P_{\mathcal N} M_z |_{\mathcal N}$. …
Real spectral shift functions for pairs of contractions and pairs of dissipative operators
本文研究了收缩算子对和耗散算子对的实谱移函数条件，推导了其存在性，并应用于耗散Schrödinger算子。
Compressions of selfadjoint and maximal dissipative extensions of non-densely defined symmetric operators
该文针对无穷维可分Hilbert空间中的非稠定对称算子，研究其自伴与最大耗散延拓在定义域闭包子空间上的压缩，重点关注定义域闭包余维数为无穷的情况，得到了这类算子特征函数的新性质。
Real-valued spectral shift functions for contractions and dissipative operators
本文针对收缩算子对与耗散算子对，公布了可积实值谱移函数存在性的充分条件，补充了此前迹类差收缩算子可积谱移函数存在性相关研究的实值性结论。
In this note we investigate complete non-selfadjointness for all maximally dissipative extensions of a Schrödinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. We show that all maximally dissipative extensions that preserve the differential expression are completely non-selfadjoint. However, it is possible for maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. We give a characterisation of these extensions and the corresponding subspaces and present a specific example.
L-resolvents of symmetric linear relations in Pontryagin spaces
本文在Pontryagin空间框架下，建立具有不合适规范的对称线性关系的边界三元组理论与Krein–Saakyan L-预解矩阵理论的联系，推广L-预解矩阵的边界算子表示公式，并应用于典范系统生成的极小线性关系。
For Maxwell operators $(E,H) \to (i ε^{-1} \nabla\times H, -i μ^{-1} \nabla \times E)$ in Lipschitz domains, we describe all m-dissipative boundary conditions and apply this result to generalized impedance and Leontovich boundary conditions including the cases of singular, degenerate, and randomized impedance coefficients. To this end we construct Riesz bases in the trace spaces associated with the curl-operator and introduce a modified version of boundary triple adapted for the specifics of Maxwell equations, namely, to the mixed-order duality of the related trace spaces. This provides a tran…
Maximally dissipative and self-adjoint extensions of $K$-invariant operators
该研究提出希尔伯特空间中$K$-不变算子的概念，推导其自伴与最大耗散扩张保持$K$-不变的条件，应用于Sturm–Liouville算子并刻画相关扩张，给出半直线上的Bessel型Schrödinger算子示例。
In this article we develop a functional model for a general maximal dissipative operator. We construct the selfadjoint dilation of such operators. Unlike previous functional models, our model is given explicitly in terms of parameters of the original operator, making it more useful in concrete applications. For our construction we introduce an abstract framework for working with a maximal dissipative operator and its anti-dissipative adjoint and make use of the Straus characteristic function in our setting. Explicit formulae are given for the selfadjoint dilation, its resolvent, a core and the…
The theory of symmetric, non-selfadjoint operators has several deep applications to the complex function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators such as Schrodinger operators in mathematical physics. Examples of simple symmetric operators include multiplication operators on various spaces of analytic functions such as model subspaces of Hardy spaces, deBranges-Rovnyak spaces and Herglotz spaces, ordinary differential operators (including Schrodinger operators from quantum mechanics), Toeplitz operator…
In the first section we provide a solution to the M. G. Krein problem about an inner description of the space $L_2(Σ,H).$ In the second section we introduce the multiplicity function for an operator measure. Making use of the description of the space $L_2(Σ,H)$ we establish the correctness of the definition and give a criterion for a spectral measure to be a dilation of a given operator measure. In the third section we prove that the set of principal vectors of an operator measure is an everywhere dense $G_δ.$ This implies, in particular, that there are a lot of principal vectors in any cyclic…
The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted $C_0$-space on the real line. A theorem of L. de Branges characterizes non--density by existence of an entire function of Krein class being related with the weight in a certain way. An analogous result holds true for weighted sup--norm approximation by entire functions of exponential type at most $τ$ and bounded on the real axis ($τ>0$ fixed). We consider approximation in weighted $C_0$-spaces by functions belonging to a prescribed subspace of entire functions which is s…
We extend the concept of Lifshits--Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Our main result is the following. Let \cH_0 and \cH_1 be separable Hilbert spaces, A_0 a self-adjoint operator in \cH_0, A_1 a self-adjoint operator in \cH_1, and B_{ij} a bounded operator from \cH_j to \cH_i, i=0,1, j=1-i, and B_10=B_01^*. Assume that the block operator matrix \bH=\bA+\bB=(A_0 & B_01 B_{10}& A_1) has reducing graph subspaces of the form {x_i\oplus Q_ji x_i: x_i\in\cH_i}, i=0,1…
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