On existence of maximal semidefinite invariant subspaces for $J$-dissipative operators

非稠定对称算子的自伴延拓与最大耗散延拓的压缩

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。

Complete Non-Selfadjointness for Schrödinger Operators on the Semi-Axis

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.

On Second-order Conditions for Quasiconvexity and Pseudoconvexity of $\mathcal{C}^{1,1}$-smooth Functions

For a $\mathcal{C}^2$-smooth function on a finite-dimensional space, a necessary condition for its quasiconvexity is the positive semidefiniteness of its Hessian matrix on the subspace orthogonal to its gradient, whereas a sufficient condition for its strict pseudoconvexity is the positive definiteness of its Hessian matrix on the subspace orthogonal to its gradient. Our aim in this paper is to extend those conditions for $\mathcal{C}^{1,1}$-smooth functions by using the Fréchet and Mordukhovich second-order subdifferentials.

The functional model for maximal dissipative operators: An approach in the spirit of operator knots

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…

Maximal lower bounds in the Löwner order

We show that the set of maximal lower bounds of two symmetric matrices with respect to the Löwner order can be identified to the quotient set $O(p,q)/(O(p)\times O(q))$. Here, $(p,q)$ denotes the inertia of the difference of the two matrices, $O(p)$ is the $p$-th orthogonal group, and $O(p,q)$ is the indefinite orthogonal group arising from a quadratic form with inertia $(p,q)$. We also show that a similar result holds for positive semidefinite maximal lower bounds with maximal rank of two positive semidefinite matrices. We exhibit a correspondence between the maximal lower bounds $C$ of two m…

A lifting theorem for 3-isometries

An operator T on Hilbert space is a 3-isometry if there exists operators B and D such that (T*)^n T^n = I+nB +n^2 D. An operator J is a Jordan operator if it the sum of a unitary U and nilpotent N of order two which commute. If T is a 3-isometry and c>0, then I-c^{-2} D + sB + s^2D is positive semidefinite for all real s if and only if T is the restriction to an invariant subspace of a Jordan operator J=U+N with the norm of N at most c. As a corollary, an analogous result for 3-symmetric operators, due to Helton and Agler, is recovered.