# -Hyponormal Operators

## Top PDF -Hyponormal Operators: ### Extensions of the results on powers of hyponormal and hyponormal operators

Firstly, we will show the following extension of the results on powers of p-hyponormal and log-hyponormal operators: let n and m be positive integers, if T is p-hyponormal for p ∈ (0, 2], then: (i) in case m ≥ p, (T n+m ∗ T n+m ) (n+p)/(n+m) ≥ (T n ∗ T n ) (n+p)/n and (T n T n ∗ ### On some fuzzy hyponormal operators

In section 2 we provide some preliminary definitions, the- orems and results, which are used in this paper. In section 3 we establish some theorems on fuzzy hyponormal operators and we introduce the concept of fuzzy class (N) operator and some properties of fuzzy class operators have been studied. ### Almost 𝜶 Hyponormal Operators with Weyl Spectrum of Area Zero

Research Article Almost α-Hyponormal Operators with Weyl Spectrum of Area Zero Vasile Lauric Department of Mathematics, Florida A&M University, Tallahassee, FL 32307, USA Correspondence [r] ### Fuglede–Putnam type theorems for \((p,k)\) quasihyponormal operators via hyponormal operators

A part of an operator is its restriction to a closed invariant subspace. A class of operators is called hereditary if each part of an operator in the class also belongs to the class. Remark 1.6 It is well known that the class FP(p-H) includes many classes of operators, such as dominant operators [11–13, 18], (p, k)-quasihyponormal operators with reducing kernels [15, 17], and w-hyponormal operators with reducing kernels . Moreover, it is known that the classes above also belong to the class of hereditary FP(H) (denote this class by HFP(H)), that is, every restriction of an operator to its closed invariant subspace also belongs to the class. See [1, 7, 13, 16, 18]. ### Some remarks on the invariant subspace problem for hyponormal operators

A hyponormal operator is called pure if it does not have a nonzero reducing sub- space to which its restriction is a normal operator. Obviously an operator in H( Ᏼ ) without a n.i.s. is pure. The following result of Putnam  leads to another reduction of the ISP for hyponormal operators. ### Fractional powers of hyponormal operators of Putnam type

As a special case, with α = 1/2 in (1.3), we recover the well-known square root prob- lem of Kato for the algebraic sum of hyponormal operators of Putnam type. Notice that similar investigations were made by the author regarding the square root problem of Kato for the algebraic sum of normal and m-sectorial operators, see, for example, [3, 4, 5, 6]. 2. Preliminaries ### 3. Some properties of paranormal and hyponormal operators

Proof. Let T ∈ (M, k), for k ≥ 2, it means that T ∈ (M, 2). One side of proposition follows directly from definition of classes (M, k). Let us prove the other side of the proposition. Consider that operator T is quasi-hyponormal. We will construct our proof following mathematical induction. For k = 2, it follows from definition of quasi-hyponormal operators. Let us consider that our claim is valid for k = n and we will prove it for k = n + 1. We have: ### Weyl type theorems and k quasi M hyponormal operators

tained an improvement of Stampﬂi’s result to p-hyponormal operators or log-hyponormal operators. Furthermore, Ch¯o and Han extended it to M-hyponormal operators as follows. Proposition . [, Theorem ] Let T be an M-hyponormal operator, and let λ be an isolated point of σ(T ). If E is the Riesz idempotent for λ, then E is self-adjoint, and R(E) = N(T – λ) = N(T – λ) ∗ . ### A Note on Hyponormal Operators

each α ∈ 0, 1, we obtain H α 0 H ⊇ H β 0 H when α ≤ β. However, the similar inclusions for the classes N α p H and H α p H are less obvious. In this section, we will examine various inclusions between these classes of operators. 1 of Theorem 2.1 has been already shown in 1. But we will give a proof for the readers’ convenience. ### Absolute continuity and hyponormal operators

ergodic theory, unitary dilation theory, and minimal normal extensions of subnormal contractions.".. KEY WORDS AND PHRASES.[r] ### Quasinilpotent Part of w Hyponormal Operators

h T = . From the spectral mapping theorem it easily follows that the spectrum of analgebraic operator is a finite set. A nilpotent operator is a trivial example of an algebraic operator. Also finite rank operators K are algebraic; more generally, if K n is a finite rank operator for some n∈ N then K is algebraic. Clearly, if T is algebraic then its dual T ∗ is algebraic. ### Results On A-Unitary, A-Normal and A-Hyponormal Operators

In this research thesis Hilbert spaces or subspaces will be denoted by capital letters, etc and , , , denote bounded linear operators where an operator means a bounded linear transformation. will denote the bounded linear operators on a complex separable Hilbert space . denotes the set of bounded linear transformations from to , which is equipped with the (induced uniform) norm.The following definitions are of essence: ### Essential spectra of quasisimilar quasihyponormal operators

whether quasisimilar hyponormal operators also have essential spectra. Later Williams (see [11, Theorem 1], [12, Theorem 3]) showed that two quasisimilar quasinormal op- erators and under certain conditions two quasisimilar hyponormal operators have equal essential spectra. Gupta [4, Theorem 4] showed that biquasitriangular and quasisimi- lar k-quasihyponormal operators have equal essential spectra. On the other hand, Yang [13, Theorem 2.10] proved that quasisimilar M-hyponormal operators have equal es- sential spectra, and Yingbin and Zikun [14, Corollary 12] showed that quasisimilar p - hyponormal operators have also equal spectra and essential spectra. Very recently, Jeon et al. [8, Theorem 5] showed that quasisimilar injective p-quasihyponormal operators have equal spectra and essential spectra. In this paper we give some conditions for operators A and B (A is left-Fredholm and B is right-Fredholm) to exist an operator C such that M C ### Vol 3, No 2 (2012)

In , the class of log - hyponormal operators is defined as follows: T is called log - hyponormal if it is invertible and satisfies log ( ) T T * p ≥ log ( ) TT * p . Class of p - hyponormal operators and class of log hyponormal operators were defined as extension class of hyponormal operators, i.e., T T * ≥ TT * . It is well known that every p - hyponormal operator is a q - hyponormal operator for p ≥ > q 0 , by the Löwner - Heinz theorem " A ≥ ≥ B 0 ensures A α ≥ B α for any α ∈ [0,1] ", and every invertible p - hyponormal operator is a log - hyponormal operator since log ( ) ⋅ is an operator monotone function. An operator T is called paranormal if Tx 2 ≤ T x 2 x for all ### Some Estimates of Certain Subnormal and Hyponormal Derivations

In this section, we approach the same problem, but in the case in which A S, B T are hyponormal operators and the Hilbert-Schmidt class is replaced with an arbitrary norm ideal. For a hyponormal operator T ∈ LH, the analytic functional calculus can be extended to a class A α σT of “pseudo-analytic” functions on σT that satisfy a certain growth condition at the boundary. ### On ∗ class A contractions

T = r(T ), the spectral radius of T). In order to discuss the relations between paranormal and p-hyponormal and log-hyponormal operators (T is invertible and log T ∗ T ≥ log TT ∗ ), Furuta et al. [] introduced a very interesting class of operators: class A deﬁned by | T  | – | T |  ≥ , where | T | = (T ∗ T )   which is called the absolute value of T ; and they ### An extension to Fuglede-Putnam’s theorem and orthogonality

These classes are interesting and have similar properties to those of hyponormal operators. The familiar Fuglede-Putnam’s theorem asserts that if A ∈ B( H ) and B ∈ B( K ) are normal operators and AX = X B for some operators X ∈ B( K , H ), then A ∗ X = X B ∗ . Many authors have extended this theorem for several classes of operators. H.I. Kim  proved that Fuglede- Putnam’s theorem holds for injective (p, k)−quasihyponormal and p−hyponormal operators. ### Spectrum of Class Operators

Tanahashi, “Isolated point of spectrum of p-hyponormal, log-hyponormal operators,” Integral Equations and Operator Theory, vol.. Yamazaki, “An operator transform from class A to the clas[r] ### Weighted boundedness for Toeplitz type operators related to strongly singular integral operators

( < p < ∞) spaces are obtained. In [, ], some Toeplitz type operators related to the sin- gular integral operators and strongly singular integral operators are introduced, and the boundedness for the operators generated by BMO and Lipschitz functions is obtained. In this paper, we will study the Toeplitz type operators related to the strongly singular integral operator and the weighted Lipschitz functions. ### Review on Overview of Oracle Operators

Oracle allows arithmetic operators to be used while viewing records from a table or while performing Data Manipulation operations such as insert, update and delete. You can use an arithmetic operator with one or two arguments to negate, add, subtract, multiply and divide numeric values. Some of these operators are also used in date time and interval arithmetic. The arguments to the operators must resolve to numeric data types or to any data types that can be implicitly converted to a numeric data types. Unary arithmetic operators return the same data type as the numeric data type of the arguments. For binary arithmetic operators, Oracle determines the arguments with the highest numeric precedence, implicitly converts the remaining arguments to that data type, and return that data type.