The k-quasi-∗-class contractions have property PF (original) (raw)
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ON k-QUASI-CLASS A CONTRACTIONS
Korean Journal of Mathematics, 2014
A bounded linear Hilbert space operator T is said to be k-quasi-class A operator if it satisfy the operator inequality T * k |T 2 |T k ≥ T * k |T | 2 T k for a non-negative integer k. It is proved that if T is a k-quasi-class A contraction, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator D = T * k (|T 2 | − |T | 2)T k is strongly stable.
Linear Algebra and its Applications, 2012
Let QA denote the class of bounded linear Hilbert space operators T which satisfy the operator inequality T * |T 2 |T ≥ T * |T| 2 T. It is proved that if T ∈ QA is a contraction, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator D = T * (|T 2 | − |T| 2)T is strongly stable. It is shown that if T ∈ QA is a contraction with Hilbert-Schmidt defect operator such that T −1 (0) ⊆ T * −1 (0), then T is completely non-normal if and only if T ∈ C 10 , and a commutativity theorem is proved for contractions T ∈ QA. Let T u and T c denote the unitary part and the cnu part of a contraction T, respectively. We prove that if A = A u ⊕ A c and B = B u ⊕ B c are QA-contractions such that μ A c < ∞, then A and B are quasi-similar if and only A u and B u are unitarily equivalent and A c and B c are quasi-similar.
SOME PROPERTIES AND CONTRACTIONS OF CLASS A(k) OPERATORS
2017
First, we give some properties of class A(k) operators which are defined in [21]. Exactly we show that if Tn is a compact operator from class A(k), then it follows that T is compact too. We introduce the class M − A(k) of operators and show some properties of this class of operators and their relationship with other classes. We also prove that if Pμ is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a class A(k) operator T where k ∈ (0, 1], then Pμ is self-adjoint, if and only if, ker(T −μ) ⊆ ker(T ∗ − μ). Second, we prove that a class A(k) of operator, where k ∈ (0, 1], is polaroid and a−Weyl’s theorem holds for class A(k) of operators. Finally, we see if T is a contraction of the class A(k) operator for k > 0, then the nonnegative operator D = ( T ∗|T |T ) 1 k+1 − |T | is a contraction whose power sequence {D}n=1 convergences strongly to a projection P and TP = O. Also, we prove if T ∈ A(k) is a contraction for 0 < k ≤ 1, then T has a Wold−type decompo...
On ∗-paranormal contractions and properties for ∗-class A operators
Linear Algebra and its Applications, 2012
C .0-contraction Tensor product An operator T ∈ B(H) is called a *-class A operator if |T 2 | |T * | 2 , and T is said to be *-paranormal if T * x 2 T 2 x for every unit vector x in H. In this paper, we show that *-paranormal contractions are the direct sum of a unitary and a C .0 completely non-unitary contraction. Also, we consider the tensor products of *-class A operators.
Some Invariant subspaces for A-contractions and applications
2006
Some invariant subspaces for the operators A and T acting on a Hilbert space H and satisfying T*AT = A and A = 0, are presented. Especially, the largest invariant subspace for A and T on which the equality T* AT = A occurs, is studied in connections to others invariant or reducing subspaces for A, or T. Such subspaces are related to the asymptotic form of the subspace quoted above, this form being obtained using the operator limit of the sequence {T*nATn; n = 1}. More complete results are given in the case when AT = A1/2TA1/2. Also, several applications for quasinormal operators are derived, involving their unitary, isometric and quasi-isometric parts, as well as their asymptotic behaviour.
Annals of the Alexandru Ioan Cuza University - Mathematics, 2013
Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let A, B be operators in B(H). In this paper we prove that if A is quasi-class A and B * is invertible quasi-class A and AX = XB, for some X ∈ C2 (the class of Hilbert-Schmidt operators on H), then A * X = XB * . We also prove that if A is a quasi-class A operator and f is an analytic function on a neighborhood of the spectrum of A, then f (A) satisfies generalized Weyl's theorem. Other related results are also given.
Properties of absolute-*-k-paranormal operators and contractions for *-A(k) operators
Studia Universitatis Babes-Bolyai Matematica
First, we see if T is absolute-*-k-paranormal for k ≥ 1, then T is a normaloid operator. We also see some properties of absolute-*-k-paranormal operator and *-A(k) operator. Then, we will prove the spectrum continuity of the class *-A(k) operator for k > 0. Moreover, it is proved that if T is a contraction of the class *-A(k) for k > 0, then either T has a nontrivial invariant subspace or T is a proper contraction, and the nonnegative operator
Let T be a k-quasi-*-class A operator on a complex Hilbert space H if 2 * 2 * 0 k k T T T T where k is a natural number. In this paper, we prove Browder's, Generalised Browder's, a-Browder's, and Generalised a-Browder's theorem for k-quasi-*-class A operators.
ON k− QUASI CLASS A n OPERATORS
In this paper, we introduce a new class of operators, called k−quasi class A * n operators, which is a superclass of hyponormal operators and a subclass of (n, k)−quasi− * −paranormal operators. We will show basic structural properties and some spectral properties of this class of operators. We show that, if T is k−quasi class A * n then σ jp (T) \ {0} = σp(T) \ {0}, σ ja (T) \ {0} = σa(T) \ {0} and T − λ has finite ascent for all λ ∈ C. Also, we will prove Browder's theorem and a−Browders theorem for k−quasi class A * n operator. 2010 Mathematics Subject Classification. 47B20, 47B37. Key words and phrases. k−quasi class A * n operator, SVEP, Ascent, Browder's theorem, a−Browders theorem.