### Some versions of Anderson's and Maher's inequalities. I.

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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.

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 =...

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 ${T}^{*k}\left(\right|T\xb2|-|T*|\xb2){T}^{k}\ge 0$ 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.

In this paper we obtain some results concerning the set $\mathcal{M}=\cup \left\{\overline{R\left({\delta}_{A}\right)}\cap {\left\{A\right\}}^{\text{'}}\phantom{\rule{0.222222em}{0ex}}A\in \mathcal{L}\left(H\right)\right\}$, where $\overline{R\left({\delta}_{A}\right)}$ is the closure in the norm topology of the range of the inner derivation ${\delta}_{A}$ defined by ${\delta}_{A}\left(X\right)=AX-XA.$ Here $\mathscr{H}$ stands for a Hilbert space and we prove that every compact operator in ${\overline{R\left({\delta}_{A}\right)}}^{w}\cap {\left\{{A}^{*}\right\}}^{\text{'}}$ is quasinilpotent if $A$ is dominant, where ${\overline{R\left({\delta}_{A}\right)}}^{w}$ is the closure of the range of ${\delta}_{A}$ in the weak topology.

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