## Banach Journal of Mathematical Analysis

- Banach J. Math. Anal.
- Volume 12, Number 3 (2018), 751-771.

### Reflexive sets of operators

Janko Bračič, Cristina Diogo, and Michal Zajac

#### Abstract

For a set $\mathcal{M}$ of operators on a complex Banach space $\mathcal{X}$, the reflexive cover of $\mathcal{M}$ is the set $Ref\left(\mathcal{M}\right)$ of all those operators $T$ satisfying $Tx\in \overline{\mathcal{M}x}$ for every $x\in \mathcal{X}$. Set $\mathcal{M}$ is reflexive if $Ref\left(\mathcal{M}\right)=\mathcal{M}$. The notion is well known, especially for Banach algebras or closed spaces of operators, because it is related to the problem of invariant subspaces. We study reflexivity for general sets of operators. We are interested in how the reflexive cover behaves towards basic operations between sets of operators. It is easily seen that the intersection of an arbitrary family of reflexive sets is reflexive, as well. However this does not hold for unions, since the union of two reflexive sets of operators is not necessarily a reflexive set. We give some sufficient conditions under which the union of reflexive sets is reflexive. We explore how the reflexive cover of the sum (resp., the product) of two sets is related to the reflexive covers of summands (resp., factors). We also study the relation between reflexivity and convexity, with special interest in the question: under which conditions is the convex hull of a reflexive set reflexive?

#### Article information

**Source**

Banach J. Math. Anal., Volume 12, Number 3 (2018), 751-771.

**Dates**

Received: 21 September 2017

Accepted: 24 January 2018

First available in Project Euclid: 18 May 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bjma/1526630423

**Digital Object Identifier**

doi:10.1215/17358787-2018-0002

**Mathematical Reviews number (MathSciNet)**

MR3824750

**Zentralblatt MATH identifier**

06946080

**Subjects**

Primary: 47A99: None of the above, but in this section

Secondary: 47L05: Linear spaces of operators [See also 46A32 and 46B28] 47L07: Convex sets and cones of operators [See also 46A55]

**Keywords**

reflexive set of operators convex set of operators locally linearly dependent operators

#### Citation

Bračič, Janko; Diogo, Cristina; Zajac, Michal. Reflexive sets of operators. Banach J. Math. Anal. 12 (2018), no. 3, 751--771. doi:10.1215/17358787-2018-0002. https://projecteuclid.org/euclid.bjma/1526630423