On unbounded hyponormal operators II

Integral Equations and Operator Theory, 1997

2022

A closed densely defined operator T on a Hilbert space H is callled M-hyponormal if D(T) ⊂ D(T *) and there exists M > 0 for which (T − zI) * x ≤ M (T − zI)x for all z ∈ C and for all x ∈ D(T). In this paper, we prove that if bounded linear operator A : H → K is such that AB * ⊆ T A, where B is a closed subnormal (resp. a closed M-hyponormal) on H, T is a closed M-hyponormal (resp. a closed subnormal) on H, then (i) AB ⊆ T * A, (ii) ran(A *) reduces B to the normal operator B| ran(A *) , and (iii) ran(A) reduces T to the normal operator T | ran(A). T (x 1 , x 2 , x 3 ......) = (2x 1 , 3x 2 , 4x 3 , 5x 4 , 6x 5 ......

2010

Let A, B * ∈ B(H) be w-hyponormal operators, and let d AB ∈ B(B(H)) denote either the generalized derivation δ AB (X) = AX -XB or the length two elementary operator AB (X) = AXB -X. We prove that d AB has the single-valued extension property, and the ) denote the space of functions which are analytic on σ (d AB ), and let H c (σ (d AB )) denote the space of f ∈ H(σ (d AB )) which are non-constant on every connected component of σ (d AB ). It is proved that, for every h ∈ H(σ (d AB )) and f , g ∈ H c (σ (d AB )), the complement of the Weyl spectrum σ w (h )) consists of isolated points in σ (h(d f (A)g(B) )) which are eigenvalues of finite multiplicity.

Glasgow Mathematical Journal, 1998

Let B(H) denote the algebra of operators (i.e., bounded linear transformations) on the Hilbert space H. A ∈ B (H) is said to be p-hyponormal (0<p<l), if (AA*)γ < (A*A)p. (Of course, a l-hyponormal operator is hyponormal.) The p-hyponormal property is monotonic decreasing in p and a p-hyponormal operator is q-hyponormal operator for all 0<q <p. Let A have the polar decomposition A = U |A|, where U is a partial isometry and |A| denotes the (unique) positive square root of A*A.If A has equal defect and nullity, then the partial isometry U may be taken to be unitary. Let ℋU(p) denote the class of p -hyponormal operators for which U in A = U |A| is unitary. ℋU(l/2) operators were introduced by Xia and ℋU(p) operators for a general 0<p<1 were first considered by Aluthge (see [1,14]); ℋU(p) operators have since been considered by a number of authors (see [3, 4, 5, 9, 10] and the references cited in these papers). Generally speaking, ℋU(p) operators have spectral proper...

1987

complete unitary invariants for a pure hyponormal operator are described and related to already known invariants. Among them an operator valued distribution supported by the spectrum has a distinguished position. The inverse problems and the relations between these invariants are also discussed. Parts of this paper were motivated by the recent work of K. Clancey.

Taiwanese Journal of Mathematics, 1997

This paper is concerned with two cosine-function-related functions which are called cosine step response and cosine cumulative output. We study some of their properties, such as measurability, continuity, infinitesimal operator, compactness, positivity, almost periodicity, and asymptotic behavior.

Mathematical Notes, 1992

In fact, let s~(x)= max a~i(x). We may assume that n = 2 p. t~<k~<n i=1 /k (x) = ~=Ji~l ai~ (z), We define the orthonormal system {~.

1991

Studia Mathematica, 2004

Weyl type theorems for p-hyponormal and M-hyponormal operators by Xiaohong Cao (Xi'an), Maozheng Guo (Beijing) and Bin Meng (Beijing) Abstract. "Generalized Weyl's theorem holds" for an operator when the complement in the spectrum of the B-Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues; and "generalized a-Weyl's theorem holds" for an operator when the complement in the approximate point spectrum of the semi-B-essential approximate point spectrum coincides with the isolated points of the approximate point spectrum which are eigenvalues. If T or T * is p-hyponormal or M-hyponormal then for every f ∈ H(σ(T)), generalized Weyl's theorem holds for f (T), so Weyl's theorem holds for f (T), where H(σ(T)) denotes the set of all analytic functions on an open neighborhood of σ(T). Moreover, if T * is p-hyponormal or M-hyponormal then for every f ∈ H(σ(T)), generalized a-Weyl's theorem holds for f (T) and hence a-Weyl's theorem holds for f (T). a (T) = C \ σ a (T). An operator T ∈ B(H) is called Fredholm if it has closed finite-codimensional range and finite-dimensional null space. The index of a Fredholm operator T ∈ B(H) is given by ind(T) = n(T) − d(T). An operator T ∈ B(H) is called Weyl if it is Fredholm of index zero, and Browder if it is Fredholm of finite ascent and descent, or equivalently, if T is Fredholm and T − λI is invertible for all sufficiently small λ = 0 in C. For T ∈ B(H), we write α(T) for the ascent of T and β(T) for the descent of T .

Advances in Pure Mathematics, 2012

T B H  to be * p-paranormal and the monotonicity of A p q. We also present an alternative proof of a result of M. Fujii, et al. [1, Theorem 3.4].

Journal of the Korean Mathematical Society, 2016

In this paper, we introduce the class of analytic extensions of M-hyponormal operators and we study various properties of this class. We also use a special Sobolev space to show that every analytic extension of an M-hyponormal operator T is subscalar of order 2k + 2. Finally we obtain that an analytic extension of an M-hyponormal operator satisfies Weyl's theorem.

Mathematische Nachrichten, 2014

In this paper we deal with the hyponormality of Toeplitz operators with matrixvalued symbols. The aim of this paper is to provide a tractable criterion for the hyponormality of bounded-type Toeplitz operators TΦ (i.e., the symbol Φ ∈ L ∞ Mn is a matrix-valued function such that Φ and Φ * are of bounded type). In particular, we get a much simpler criterion for the hyponormality of TΦ when the co-analytic part of the symbol Φ is a left divisor of the analytic part.

2015

This thesis presents a version of the spectral theorem for unbounded self-adjoint operators on a Hilbert space H. Given such an operator (A, D A), the resolvent R A (z) gives rise to the quadratic form q R A (z) (u), which is a Herglotz function in z for each u ∈ H. This corresponds to a Borel transform of some nite Borel measure µ u. Using Stieltjes' inversion formula, µ u can be completely determined for each u ∈ H. Conversely, any projection-valued measure dened on H gives a self-adjoint (A, D A) and associates q A (u) to a measure µ u for each u ∈ H. The classes of measure {µ u } u∈H and {µ u } u∈H are the same and dene a unique representation of (A, D A). Finally, {µ u } u∈H is combined to a spectral measure µ in the spectral decomposition of (A, D A). This theory will be illustrated by analyzing the Sturm-Liouville operators. The number of self-adjoint extensions, a resolvent formula, a spectral mapping will be discussed. The dependence between q(x) and how a Sturm-Liouville operator behaves at the boundary will be investigated. I want to thank Erik Wahlén for his enthusiasm and his most patient guidance through this work. Sammanfattning Detta arbete presenterar en version av spektralsatsen för självadjungerande obegränsade operatorer. Om (A, D A) är en sådan operator på ett Hilbertrum H, ger dess resolvent R A (z) upphov till en kvadratisk form q R A (z) (u) som är en Herglotz funktion för varje xt u ∈ H. Detta motsvarar en Boreltransform för något ändligt Borelmått µ u. Med hjälp av Stieltjes inversionsformell kan måttet µ u uttryckas explicit. Omvänt denierar ett projektionsvärt mått på H en självadjungerande operator (A, D A). Måttet associerar q A (u) till ett ändligt Borelmått µ u. De två klasserna av mått {µ u } u∈H och {µ u } u∈H är lika med varandra. Måtten µ u denierar en unik representation av (A, D A). Slutligen förenas måtten µ u till ett spektralmått µ i diagonaliseringen av (A, D A). Teorin illustreras i en analys av Sturm-Liouville operatorer. Antal självadjungerande utvidgningar, en resolventformel samt en spektralavbildning undersöks. Kopplingen mellan operatorns beteenden nära randen och funktionen q(x) också klargörs. Jag vill tacka Erik Wahlén för den entusiasmerande och den tålmodigaste handledningen genom detta arbete. Populärvetenskaplig sammanfattning Många dierentialekvationer inom fysik och matematik är kopplade till obegränsade självadjungerande operatorer. Vi skulle vilja formulera en version av spektralsatsen för dessa operatorer. Denna sats är ett känt verktyg från lineär algebra för att förenkla symmetriska matriser. Dock kan vi inte studera obegränsade operatorer med hjälp av de gamla denitionerna och resultaten rakt av. Vi behöver en ny allmännare denition av operatorer och nya resultat. I detta arbete presenterar vi en teori anpassad för obegränsade självadjungerande operatorer. Därefter bevisas spektralsatsen för dessa och avslutningsvis illustreras teorin i en analys av Sturm-Liouville operatorer, som förekommer i t ex Schrödingers tidsoberoende vågekvation.

Proceedings of the American Mathematical Society, 1981

If AX = XB* with A and B A/-hyponormal, then A*X = XB. Furthermore, (ran X)~ reduces A, ker X reduces B, and /4|(ran X)~ and 2?*|kerx X are unitarily equivalent normal operators. An asymptotic version is also proved.

Afrika Matematika, 2020

Let T andT be closed linear operators in a Hilbert space. Let the spectrum of T is in the right half-plane. We suggest the conditions, under which the spectrum ofT also lies in the right half-plane.