Aleksej Turnšek

Elementary operators and orthogonality

Abstract Let H be a separable infinite dimensional Hilbert space and let B ( H ) denote the algebra of operators on H into itself. We study the elementary operators φ, φ * : B ( H )→ B ( H ) defined by φ(X)=∑ i=1 k A i XB i and φ * (X)=∑ i=1 k A i * XB i * . We prove that: (i) when ∥φ∥⩽1 , then ∥φ(X)−X+S∥⩾∥S∥ for all X∈ B ( H ) and all S∈ ker φ ; (ii) when ∑ i=1 k A i A i * ⩽1 , ∑ i=1 k A i * A i ⩽1 , ∑ i=1 k B i B i * ⩽1 and ∑ i=1 k B i * B i ⩽1 , then for S∈ ker φ∩ ker φ * ∩ C p (the von Neumann–Schatten p-class), 1⩽p , ∥φ(X)−X+S∥ p ⩾∥S∥ p and ∥φ * (X)−X+S∥ p ⩾∥S∥ p for all X∈ B ( H ) , where by convention ∥Y∥ p =∞ if Y∉ C p ; (iii) let (M i ) i=1 k and (N i ) i=1 k be separately commuting sequences of normal operators and let Δ: B ( H )→ B ( H ) be defined by Δ(X)=∑ i=1 k M i XN i . If Δ(X)∈ C 2 and S∈ ker Δ∩ C 2 , then ∥Δ(X)+S∥ 2 2 =∥Δ(X)∥ 2 2 +∥S∥ 2 2 .

A quantitative version of Herstein’s theorem for Jordan -isomorphisms

We study linear mappings between -algebras and , which approximately satisfy Jordan multiplicativity condition and a -preserving condition (that is, the so-called -approximate Jordan -homomorphisms). We first prove that every such mapping is automatically continuous and we give the estimates of its norm, as well as the estimates of the norm of its inverse if it is bijective. If , , and is a bijective -approximate Jordan -homomorphism with sufficiently small , then either has a large norm, or is close to a Jordan -isomorphism, that is, to a mapping of the form , or , for some unitary . We also give the corresponding quantitative estimate.

Circular two-sided multiplications

In this note we consider circular and strongly circular two-sided multiplications acting on or on minimal norm ideals of . We prove that strong circularity of implies circularity of A or of B. If A and B are irreducible and is acting on some minimal norm ideal different from the Hilbert-Schmidt class, then is strongly circular if and only if A or B is strongly circular.