A perturbed elementary operator and range-kernel orthogonality. (English) Zbl 1082.47031

Duggal, B. P.

Proc. Am. Math. Soc. 134, No. 6, 1727-1734 (2006).

Summary: Let \(B(\mathcal{H})\) denote the algebra of operators on a Hilbert \(\mathcal{H}\). If \(A_j\) and \(B_j\in B(\mathcal{H})\) are commuting normal operators, and \(C_j\) and \(D_j\in B(\mathcal{H})\) are commuting quasi-nilpotents such that \(A_jC_j-C_jA_j=B_jD_j-D_jB_j=0\), then define \(M_j, N_j\in B(\mathcal{H})\) and \({\mathcal E}, E\in B(B(\mathcal{H}))\) by \(M_j=A_j+C_j\), \(N_j=B_j+D_j\), \({\mathcal E}(X)=A_1XA_2+B_1XB_2\) and \(E(X)=M_1XM_2+N_1XN_2\). It is proved that if \(E^{-1}(0)\subseteq H_0({\mathcal E})={\mathcal E}^{-1}(0)\) and \(X\in E^{-1}(0)\), then \(|| X||\leq k\,\text{dist}(X, {\mathcal E}(B(\mathcal{H})))\), where \(k\geq 1\) is some scalar and \(H_0({\mathcal E})\) is the quasi-nilpotent part of the operator \({\mathcal E}\).

MSC:

47B47	Commutators, derivations, elementary operators, etc.
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A10	Spectrum, resolvent
47B40	Spectral operators, decomposable operators, well-bounded operators, etc.
47A30	Norms (inequalities, more than one norm, etc.) of linear operators

Keywords:

Hilbert space; normal operator; quasi-nilpotent operator; generalised scalar operator; range-kernel orthogonality; perturbation; norm