Some properties of paranormal and hyponormal operators (original) (raw)
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In this paper we introduce a new class of operators called M−quasi paranormal operators. A bounded linear operator T in a complex Hilbert space H is said to be a M−quasi paranormal operator if it satisfies ∥T 2x∥ 2 ≤ M∥T 3x∥ · ∥T x∥, ∀x ∈ H, where M is a real positive number. We prove basic properties, the structural and spectral properties of this class of operators.
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In this paper, we introduce a new class of operators, called m-quasi class A(k *) operators, which is a superclass of hyponormal operators and a subclass of absolute-(k * , m)-paranormal operators. We will show basic structural properties and some spectral properties of this class of operators. We show that if T is m-quasi class A(k *), then σnp(T) \ {0} = σp(T) \ {0}, σna(T) \ {0} = σa(T) \ {0} and T − µ has nite ascent for all µ ∈ C. Also, we consider the tensor product of m-quasi class A(k *) operators. Dedicated to the memory of Professor Takayuki Furuta with deep gratitude.
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Let T be a bounded linear operator on a complex Hilbert space H. T is called a -paranormal operator T if kTxk2 kT2xkkxk for all x 2 H. ”-paranormal” is a generalization of hyponormal (TT TT), and it is known that a -paranormal operator has several interesting properties. In this paper, we prove that if T is polynomially -paranormal, i.e., there exists a nonconstant polynomial q(z) such that q(T) is -paranormal, then T is isoloid and the spectral mapping theorem holds for the essential approximate point spectrum of T. Also, we prove related results.
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In the present paper, we prove spectral mapping theorem for (m,n)-paranormal operator T on a separable Hilbert space, that is, f (?w(T)) = ?w(f(T)) when f is an analytic function on some open neighborhood of ?(T). We also show that for (m,n)-paranormal operator T, Weyl?s theorem holds, that is, ?(T)-?w(T) = ?00(T). Moreover, if T is algebraically (m,n)-paranormal, then spectral mapping theorem and Weyl?s theorem hold.
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Journal of Inequalities and Applications, 2013
Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class of operators satisfying T * T k x 2 ≤ T k+2 x T k x for all x ∈ H, where k is a natural number. This class includes the classes of *-paranormal and k-quasi-*-class A. We prove some of the properties of these operators.