Roman Drnovšek

Triangularizing semigroups of positive operators on an atomic normed Riesz Space

In the rst part of the paper we prove several results on the ex istence of invariant closed ideals for semigroups of bounded operators on a normed Riesz space of dimension greater than possessing an atom For instance if S is a multiplicative semigroup of positive oper ators on such space that are locally quasinilpotent at the same atom then S has a non trivial invariant closed ideal Furthermore if T is a non zero positive operator that is quasinilpotent at an atom and if S is a multiplicative semigroup of positive operators such that TS ST for all S S then S and T have a common non trivial invariant closed ideal We also give a simple example of a quasinilpotent compact pos itive operator on the Banach lattice l with no non trivial invariant band The second part is devoted to the triangularizability of collections of operators on an atomic normed Riesz space L For a semigroup S of quasinilpotent order continuous positive bounded operators on L we determine a chain of invariant closed bands If in addition L has order continuous norm then this chain is maximal in the lattice of all closed subspaces of L The paper will appear in the Proceedings of the Edinburgh Mathematical Society This work was supported in part by the Research Ministry of Slovenia Introduction and Preliminaries Invariant closed subspaces of bounded operators are among the most studied topics in the operator theory In the case of positive operators on Banach lattices the order structure enable us to obtain invariant subspaces of simple geometrical forms In the paper Choi et al studied invariant closed sub spaces of semigroups of positive operators on L spaces The discrete part of these results motivated our investigations On the other hand Abramovich Aliprantis and Burkinshaw obtained some results on existence of invariant closed subspaces of an operator on an l space p which commutes with some non zero locally quasinilpotent positive operator In the present normed Riesz space setting we are extending and strengthening the results of both papers The proofs of our results mainly follow similar ideas that were introduced in and It should be noted that Jahandideh also studied collections of positive operators In he considered the positive commutant of a given positive operator and collections of positive operators dominated by a given operator In Theorem he proved a special case of our Theorem while in the rest of operators on AM spaces are studied The reader is assumed familiar with the general notions on normed Riesz spaces see the books of Luxemburg Zaanen Zaanen Schaefer Aliprantis Burkinshaw and Meyer Nieberg We shall recall some of the relevant facts Let L be a real normed Riesz space The norm of L is said to be order continuous if kf k for any downwards directed system f in L f means that infff g and for each and there exists such that f infff f g The space L is Dedekind complete if every non empty subset which is bounded from above has a supremum The cone of positive elements of L is denoted by L The positive part the negative part and the modulus or the absolute value of f L are de ned by f supff g f supf f g and jf j supff fg respectively A vector subspace I of L is said to be an order ideal if jgj jf j and f I imply that g I An ideal I is called a band if A I with supA L imply that supA I Recall that bands are necessarily closed subspaces Two elements f g L are called orthogonal or disjoint if inffjf j jgjg If A is a subset of L let A be the disjoint complement of A i e f A if and only if f is orthogonal to all elements of A An atom of L is a non zero element f L such that if g f then g f for some In other words f L is an atom i the ideal or the band generated by f is one dimensional In this case the ray Rf f f g is said to be an extreme ray A normed Riesz space L is said to be atomic if there exists an orthogonal maximal system of atoms ff g A L that is infff f g if and if infff f g for all A then f It is worth mentioning that every atomic normed Riesz space is Riesz isomorphic to an order dense Riesz subspace of IR with the coordinatewise order The word operator will be synonymous with bounded linear transfor mation A subset Z of L is invariant under an operator T on L if Tf Z for all f Z A subspace of L is non trivial if it is di erent from f g and L If C is a collection of operators on L then a subspace of L is said to be C invariant whenever it is invariant under every member of C If this subspace is non trivial we also say that C has a non trivial invariant subspace An op erator T on L is called positive if the positive cone L is invariant under T A positive operator T on L is said to be order continuous if Tf for any downwards directed system f in L In general the Banach algebra B L of all operators on L is not a Riesz space under its canonical ordering S T i the operator S T is positive We therefore say that an operator T on L has a modulus if the modulus jT j supfT Tg exists in B L If P and Q are any sets we adopt the standard convention that P Q means P Q and P Q We rst show the following result on the existence of closed ideals of a normed Riesz space Lemma Let I and J be closed ideals of a normed Riesz space L such that I J and the quotient space G J I is at least two dimensional Then there exists a closed ideal K such that I K J Proof Denote by q the canonical quotient map J G If we show that there exists a non trivial closed ideal K of G then K q K is a closed ideal satisfying the condition of the lemma Assume on the contrary that f g and G are the only closed ideals of G If g G then g or g Indeed if g and g then the closed ideal I f g generated by g is proper since I fg g L Thus g or g for each g G which imply that G is totally ordered By proposition II G is isomorphic to IR and hence one dimensional This contradiction completes the proof of the lemma A chain is a family of subspaces of L that is totally ordered by inclusion A collection C of operators on L is ideal triangularizable if there is a chain of C invariant closed ideals which is maximal in the lattice of all closed ideals of L This notion has been introduced in as a Banach lattice analogue of the well known concept of triangularizability see for example Recall that a collection C of operators on L is triangularizable if there is a chain of C invariant closed subspaces which is maximal in the lattice of all closed subspaces of L Using Lemma we prove that a maximal closed ideal chain is also a maximal closed subspace chain Proposition Let C be a chain of closed ideals of L which is maximal in the lattice of all closed ideals Then C is also maximal in the lattice of all closed subpaces of L Proof By lemma it is enough to show that the chain C is simple i e it satis es the following conditions i f g C L C ii if C is a subfamily of C then the closed subspaces

An irreducible semigroup of non-negative square-zero operators

We construct an irreducible multiplicative semigroup of non-negative square-zero operators acting onLp[0,1), for 1≤p<∞.

Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces

Relatively recently, K.M.R. Audenaert (2010), R.A. Horn and F. Zhang (2010), Z. Huang (2011), A.R. Schep (2011), A. Peperko (2012), D. Chen and Y. Zhang (2015) have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite non-negative matrices that define operators on sequence spaces. In the current paper we extend and refine several of these results and also prove some analogues for the numerical radius. Some inequalities seem to be new even in the case of

ON SEMITRANSITIVE JORDAN ALGEBRAS OF MATRICES

n\times n

An operator is a product of two quasi-nilpotent operators if and only if it is not semi-Fredholm

non-negative matrices.

A characterization of commutators of idempotents

A set

On transitive linear semigroups

\mathcal{S}

From local to global ideal-triangularizability

of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists

The von Neumann entropy and unitary equivalence of quantum states

A\in \mathcal{S}

On positive unipotent operators on Banach lattices

such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are: (1) Every irreducible semitransitive Jordan algebra is actually transitive. (2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index.