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

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.

### 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.

Zbl 1010.47013
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