Remarks on the Structure of Complex Symmetric Operators

A conjugation C is antilinear isometric involution on a complex Hilbert space , and is called complex symmetric if T* = CTC for some conjugation C. We use multiplicity theory to describe all conjugations commuting with a fixed positive operator. We expand upon a result of Garcia and Putinar to provide a factorization of complex symmetric operators which is based on the polar decomposition. This paper is based in part on the first author’s Master’s Project.     

Three Questions About Complex Symmetric Operators

We pose three questions about the structure and application of complex symmetric operators.     

Aluthge Transforms of Complex Symmetric Operators

If denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator . In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators (T iscomplex symmetric if there exists a conjugate-linear, isometric involution so that T = CT*C). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if T is complex symmetric, then and are unitarily equivalent, (3) if T is complex symmetric, then if and only if T is normal, (4) if and only if T ² = 0, and (5) every operator which satisfies T ² = 0 is necessarily complex symmetric. This work partially supported by National Science Foundation Grant DMS 0638789.     

Some Remarks on Complex Crowns of Riemannian Symmetric Spaces

We discuss several results and problems in which complex geometric constructions appear in the geometry of real pseudo Riemannian symmetric spaces in connection with analysis on such spaces.     

复对称算子的一些等价性质

证明了在复对称算子的前提下,对数-亚正规算子与正规算子是等价的,并且给出了复对称算子的一些等价性质;最后通过给出例子来说明我们的结论.     

Variational-iterative Calculations for Symmetric Operators on Real and Complex Spaces

Approximate solutions to problems from quantum scattering theoryand the theory of gauge invariance are obtained. The methodused is a variational-iterative technique applied to operatorson a complex space, not necessarily with a discrete spectrum.     

A Note on Aluthge Transforms of Complex Symmetric Operators and Applications

In this paper, we reprove that: (i) the Aluthge transform of a complex symmetric operator [(T)\tilde] = |T|^\frac12 U|T|^\frac12\tilde{T} = |T|^{\frac{1}{2}} U|T|^{\frac{1}{2}} is complex symmetric, (ii) if T is a complex symmetric operator, then ([(T)\tilde])^*(\tilde{T})^{*} and [(T^*)\tilde]\widetilde{T^{*}} are unitarily equivalent. And we also prove that: (iii) if T is a complex symmetric operator, then [((T^*))\tilde]_s,t\widetilde{(T^{*})}_{s,t} and ([(T)\tilde]_t,s)^*(\tilde{T}_{t,s})^{*} are unitarily equivalent for s, t > 0, (iv) if a complex symmetric operator T belongs to class wA(t, t), then T is normal.     

Invariant Differential Operators on Symmetric Cones and Hermitian Symmetric Spaces

We give a brief survey on the study of constructions of invariant differential operators on Riemannian symmetric spaces and of combinatorial and analytical properties of their eigenvalues, and pose some open questions.     

微分算子的对称扩张及Friedrichs扩张的辛几何刻画

本文在加权Hilbert空间L2(I,r(x))(I=(a,6),-∞≤a 0)中,利用辛几何,刻画了n阶对称微分算式的最小算子的对称扩张(含自伴扩张)及 Friedrichs扩张,分别获得了其扩张为对称扩张、Friedrichs扩张的充分必要条件．     

Generalized Resolvents of Symmetric Operators

Mathematical Notes -     

Non-Self-Adjoint Extensions of Symmetric Operators

A previous result of the author concerning almost unitary operators is applied to the spectral analysis of non-self-adjoint extensions of symmetric operators. For this purpose, the Cayley transform of such an extension is written as a perturbation of a unitary operator by a finite-rank operator of a special form in terms of the Weyl function. Bibliography: 3 titles.     

关于控制算子的若干注记

Let B(H) be the set of all bounded linear operators on a Hilbert space H. An operator T∈B(H) is called dominant if (T-λ)(T-λ)^*≤M_λ²(T-λ)^*(T-λ),?λ∈C.The numerical range of T is difined by W (T) = {(Tx, x): ‖x‖ = 1, x∈H}. In Section 1 some new characteristic of dominant operators are given. If C = AB - BA, we prove that O∈W(C)^- then A is a dominart or φ-quasihy ponor-mal. In Section 2 we prove that O∈σ_e(△_A^σ) if A is a dominant, where(?), we also prove that if A∈B(H) is a norm attaining Ф-quasihyponormal, then A has a non-trivial invariant subspace. In Section 3 we discuss the closeness of the range of bounded linear operator F_AB:X→AX-XB, and prove that R(δ_A)∩{A}′∩{Aⁿ}′=0, where δ_A:X→AX-XA.     

Some Remarks on the Boundedness of Pseudodifferential Operators

Some methods are described for reducing the problem of the boundedness of pseudodifferential operators (ΨDOs) to the theory of Fourier multipliers. Special attention is given to the boundedness of ΨDOs in Besov - Triebel - Lizorkin spaces.     

Some Remarks on Disjointly Strictly Singular Positive Operators

We characterize disjointly strictly singular positive operators by means of the compactness of their restrictions to infinite dimensional sublattices. We also give some results concerning duality in this class of operators in the setting of L^p-spaces. Mathematics Subject Classification 2000: 47B65 Partially supported by DGES BFM2001-1284.     

对称Fock空间上的加权复合算子

本文研究了可分Hilbert空间上的对称Fock空间上有界(线性和反线性)加权复合算子. 完全刻画了酉加权复合算子和自伴加权复合算子, 并考虑了一类正规加权复合算子.     

Remarks on Maximal Operators Over Arbitrary Sets of Directions

Throughout this paper, we shall let be a subset of [0, 1] havingcardinality N. We shall consider to be a set of slopes, andfor any s , we shall let e_s be the unit vector of slope s inR². Then, following [7], we define the maximal operator on R²associated with the set by The history of the bounds obtained on is quite curious. The earliest study of relatedoperators was carried out by Cordoba [2]. He obtained a boundof C(1 + log N) on the L² operator norm of the Kakeya maximaloperator over rectangles of length 1 and eccentricity N. Thisoperator is analogous to with However, for arbitrary sets, the best known result seems to be C(1 + log N). This followsfrom Lemma 5.1 in [1], but a point of view which produces aproof appears already in [8]. However, in this paper, we provethe following.