On the range and the kernel of the operator 𝑋\mapsto𝐴𝑋𝐵-𝑋

Mazouz, A.

doi:10.1090/S0002-9939-99-04754-1

On the range and the kernel of the operator $X\mapsto AXB-X$
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Proc. Amer. Math. Soc. 127 (1999), 2105-2107 Request permission

Abstract:

Let $L(H)$ denote the algebra of (bounded linear) operators on the separable complex Hilbert space $H$, and let $(\mathfrak I;\| . \|_{\mathfrak I})$ denote a norm ideal in $L(H)$. For $A,B\in L(H)$, let the derivation $\delta _{A,B}\colon L(H)\to L(H)$ be defined by $\delta _{A,B}(X)=AX-XB$, and let $\Delta _{A,B}:L(H)\to L(H)$ be defined by $\Delta _{A,B}(X)=AXB-X$. The main result of this paper is to show that if $A$, $B$ are contractions, then for every operator $T\in \mathfrak J$ such that $ATB=T$, then $\|AXB-X+T\|_{\mathfrak J}\ge \|T\|_{\mathfrak J}$ for all $X\in \mathfrak J$.

References

Joel Anderson, On normal derivations, Proc. Amer. Math. Soc. 38 (1973), 135–140. MR 312313, DOI 10.1090/S0002-9939-1973-0312313-6
Hong Ke Du, Another generalization of Anderson’s theorem, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2709–2714. MR 1273496, DOI 10.1090/S0002-9939-1995-1273496-1
B. P. Duggal, On intertwining operators, Monatsh. Math. 106 (1988), no. 2, 139–148. MR 968331, DOI 10.1007/BF01298834
B. P. Duggal, A remark on normal derivations of Hilbert-Schmidt type, Monatsh. Math. 112 (1991), no. 4, 265–270. MR 1141094, DOI 10.1007/BF01351767
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142, DOI 10.1090/mmono/018