On the range kernel orthogonality and \(p\)-symmetric operators. (English) Zbl 1112.47026

Bouali, Said; Bouhafsi, Youssef

Math. Inequal. Appl. 9, No. 3, 511-519 (2006).

Let \(Z\) be a Banach space with norm \(\|.\|\) and let \(X,Y\subseteq Z\) be two subspaces. Then \(X\) is said to be orthogonal to \(Y\) if \(\| x+ y\|\geq\| y\|\) for every \(x\in X\) and every \(y\in Y\). When \(Z\) is a Hilbert space, this is the regular orthogonality induced by the inner product.
Now consider the Banach space \(B(H)\) of all bounded linear operators on a Hilbert space \(H\). For \(A\in B(H)\), the inner derivation \(\delta_A: B(H)\to B(H)\) is defined by \(\delta_A(T)= AT- TA\), \(T\in B(H)\). A number of authors have investigated the problem for what operators \(A\), the range of \(\delta_A\) is orthogonal to its kernel, i.e., \(\|\delta_A(T)+ S\|\geq\| S\|\) for all \(T\in B(H)\) whenever \(\delta_A(S)= AS- SA= 0\). The authors of the present paper give a brief comment on this problem [cf.also B.P.Duggal, Int.J.Math.Math.Sci.27, No.9, 573–582 (2001; Zbl 1010.47013)].
Their main result is the proof of range-kernel orthogonality in the case when \(A\) is a cyclic subnormal operator and \(\|\cdot\|\) is the usual operator norm on \(B(H)\). It is shown also that the cyclicity requirement is essential – there are noncyclic subnormal operators for which the range-kernel orthogonality does not hold.

Reviewer: Khristo N. Boyadzhiev (Ada)

MSC:

47B47	Commutators, derivations, elementary operators, etc.
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A30	Norms (inequalities, more than one norm, etc.) of linear operators
47A16	Cyclic vectors, hypercyclic and chaotic operators
47B20	Subnormal operators, hyponormal operators, etc.

Keywords:

cyclic operator; operator derivation; Fuglede-Putnam theorem

Citations:

Zbl 1010.47013