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On Minimizing ||S−(AX−XB)||Pp

Mecheri, Salah — 2000

Serdica Mathematical Journal

In this paper, we minimize the map Fp (X)= ||S−(AX−XB)||Pp , where the pair (A, B) has the property (F P )Cp , S ∈ Cp , X varies such that AX − XB ∈ Cp and Cp denotes the von Neumann-Schatten class.

Fuglede-Putnam theorem for class A operators

Salah Mecheri — 2015

Colloquium Mathematicae

Let A ∈ B(H) and B ∈ B(K). We say that A and B satisfy the Fuglede-Putnam theorem if AX = XB for some X ∈ B(K,H) implies A*X = XB*. Patel et al. (2006) showed that the Fuglede-Putnam theorem holds for class A(s,t) operators with s + t < 1 and they mentioned that the case s = t = 1 is still an open problem. In the present article we give a partial positive answer to this problem. We show that if A ∈ B(H) is a class A operator with reducing kernel and B* ∈ B(K) is a class 𝓨 operator, and AX =...

Isolated points of spectrum of k-quasi-*-class A operators

Salah Mecheri — 2012

Studia Mathematica

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class, denoted *, of operators satisfying T * k ( | T ² | - | T * | ² ) T k 0 where k is a natural number, and we prove basic structural properties of these operators. Using these results, we also show that if E is the Riesz idempotent for a non-zero isolated point μ of the spectrum of T ∈ *, then E is self-adjoint and EH = ker(T-μ) = ker(T-μ)*. Some spectral properties are also presented.

Commutants and derivation ranges

Salah Mecheri — 1999

Czechoslovak Mathematical Journal

In this paper we obtain some results concerning the set = R ( δ A ) ¯ { A } ' A ( H ) , where R ( δ A ) ¯ is the closure in the norm topology of the range of the inner derivation δ A defined by δ A ( X ) = A X - X A . Here stands for a Hilbert space and we prove that every compact operator in R ( δ A ) ¯ w { A * } ' is quasinilpotent if A is dominant, where R ( δ A ) ¯ w is the closure of the range of δ A in the weak topology.

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