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 ∗

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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.

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]

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 [1]. 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].

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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 [14] leads to another reduction of the ISP for **hyponormal** **operators**.

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

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:

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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) ∗ .

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.

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ergodic theory, unitary dilation theory, and minimal normal extensions of subnormal contractions.".. KEY WORDS AND PHRASES.[r]

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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.

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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:

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

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In [23], 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

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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.

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

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 ∗ [12]. Many authors have extended this theorem for several classes of **operators**. H.I. Kim [11] proved that Fuglede- Putnam’s theorem holds for injective (p, k)−quasihyponormal and p−**hyponormal** **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]

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( < 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.

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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.

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