Ideal-triangularizability of nil-algebras generated by positive operators

A quantum measurement, often referred to as positive operator-valued measurement, is a set of positive operators P j = P j † ⩾ 0 summing to identity, ∑ j P j = 𝟙 . This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. We describe transformations in the set of quantum measurements by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity we arrive at blockwise bistochastic matrices (also called quantum magic squares). Analyzing their dynamical properties, we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in the set of quantum measurements.

Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σew(T) of the operator T is a set $$\sigma_{\rm ew}(T) = {\mathop{\cap}\limits_{K \in {\mathcal {K}}(E)}} \sigma (T + K)$$ , where $${\mathcal K}(E)$$ is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum $$\sigma_{\rm ew}^+(T) = \bigcap\limits_{0 \le K \in {\mathcal K}(E)} \sigma (T + K), \ \ \ \ \ \sigma_{\rm ew}^-(T) = \bigcap\limits_{0 \le K \in {\mathcal K}(E) \le T} \sigma (T - K)$$ are introduced in the article. The conditions by which $$r(T) \not\in \sigma_{\rm ef}(T)$$ implies either $$r(T) \not\in \sigma_{\rm ew}^+(T)$$ or $$r(T) \not\in \sigma_{\rm ew}^-(T)$$ are investigated, where σef(T) is the Fredholm essential spectrum. By this reason, the relations between coefficients of the main part of the Laurent series of the resolvent R(., T) of a positive operator T around of the point λ = r(T) are studied. The example of a positive integral operator T : L1→ L∞ which doesn’t dominate a non-zero compact operator, is adduced. Applications of results which are obtained, to the spectral theory of band irreducible operators, are given. Namely, the criteria when the operator inequalities 0 ≤ S < T imply the spectral radius inequality r(S) < r(T), are established, where T is a band irreducible abstract integral operator.

Throughout this paper we denote by L p the Banach lattice of p-integrable functions on a σ-finite measure space (X, B, μ), where 1 ≤ p ≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on L p-spaces. For general information about positive operators on Banach lattices we refer to the texts [1]. [20], and [36]. Our focus on L p-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of L p-spaces. We will discuss results related to the boundedness of positive linear operators on L p-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from L p to L q. Then we will show how this result relates to the change of density result of Weis [33]. Next the equality case of Schur’s criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on L p-spaces and discuss as an example the weighted composition operators on L p-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikisin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new.

On the ring ideal generated by a positive operator

An operator T from a Banach space E into a Banach lattice F is said to be semi-compact if for each e > 0, there exists some u ∈ F+ such that T (BE) ⊂ [−u, u] + eBF where BH is the closed unit ball of H = E or F and F+ = {x ∈ F : 0 ≤ x}. Contrarily to compact operators, the class of semi-compact operators satisfies the domination problem. Indeed, if S and T are operators from a Banach lattice E into F such that 0 ≤ S ≤ T and T is semi-compact, then S is semi-compact (Theorem 18.20 of [3]). It is clear that each compact operator is semi-compact but a semicompact operator is not necessary compact. In fact, the operator identity Idl∞ : l∞ −→ l∞ is semi-compact but it is not compact. Let us recall that Dodds and Fremlin ([11], Theorem 125.5) proved that if E and F are two Banach lattices such that the norms of F and of the topological dual E ′ are order continuous, then each semi-compact operator T from E into F which is AM-compact (i.e. a regular operator such that the image of each order bounded subset of E is a relatively compact subset of F ) is compact. In [5], we studied the compactness of positive Dunford-Pettis operators, and we gave several characterizations of Banach lattices on which each

Chapter 2 Positive operators

An effective method of investigation of positive maps on the set of positive definite operators

Let A be a *-algebra of operators on a Hilbert space H, and let Ah, A+ denote respectively the sets of self-adjoint and positive operators in A. A+ is a positive cone in Ah and it induces a partial ordering in Ah. The lattice properties of Ah were studied by R. Archbold in (1) and (2), and Chu Cho-Ho gave a different proof of some of his results in (3).

Let Z be a Banach lattice endowed with positive cone C and an order-continuous norm j.j .Let G be a left-amenable semigroup of positive linear endomorphisms of Z .Then the positive fixed points Co of Z under G form a lattice cone, and their linear span Z0 is a Banach lattice under an order-continuous norm ||.||0 which agrees with |.| on Co.A counterexample shows that under the given conditions Z0 need not contain all the fixed points of Z under G , and need not be a sublattice of (Z, C).The paper concludes with a discussion of some related results.Let G be a semigroup.We denote by m(C7) the Banach space of all bounded linear functions from G into the real numbers R, under the supremum norm.We denote by m'iG) the Banach dual of w(C7).With each T g G we associate an endomorphism Tm of /n(C7) defined by iTmb)iU) = biTU) for U G G and b G m(G) where TU denotes the composite of T and U under the semigroup operation.An element p G m*iG) is called a mean for G iff inf biT) < pib) < sup biT) for all b G G TG TG and left-invariant for G iff T'mp=p forallTGG, where T'm denotes the adjoint of Tm .Following M. Day [1, p. 108] we call the semigroup G left-amenable iff there exists a left-invariant mean for G.In particular, any Abelian semigroup is left-amenable [1, Theorem 4, p. 108].A Banach lattice is said to have order-continuous norm iff every decreasing sequence of positive elements is norm convergent [3, 5.12, p. 92; 5.10(d), p. 89].Theorem.Let (Z,C, ||.||) be a Banach lattice with order-continuous norm.Let G be a left-amenable semigroup of positive linear operators from Z into Z.Define C0 = {x G C : Tx = x for all T gG} , Z0 = C0 -CQ.

If the vector space \({\mathcal{L}}^r(E, F)\) of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for \({\mathcal{L}}^r(E, F)\) to be a vector lattice. \({\mathcal{L}}^r(E, F)\) and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces \({\mathcal{V}}(E, F)\) of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space \({\mathcal{L}}^r(E, F)\) or, in the case that \({\mathcal{L}}^r(E, F)\) is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices \({\mathcal{V}}(E, F)\), where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and \({\mathcal{V}}(E, F)\) are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in \({\mathcal{V}}(E,F)\). The paper contains also some generalization of results obtained for the case \({\mathcal{V}}(E, F) = {\mathcal{L}}^r(E, F)\) in [10].

The main aim of this paper is studying the family $$W_b(E,F)$$ of b-weakly compact operators between two Banach lattices. For an order dense sublattice G of a vector lattice E, if $$T:G\rightarrow F$$ is a b-weakly compact operator between two Banach lattices, then $$T\in W_b(E,F)$$ whenever the norm of E is order continuous and $$T:E\rightarrow F$$ is a positive operator. We also investigate the relationship between $$W_b(E,F)$$ and some other classes of operators like $$L^{(1)}_c(E,F)$$ and $$L^{(2)}_c(E,F)$$ .

We use the method of minimal vectors to prove that certain classes of positive quasinilpotent operators on Banach lattices have invariant subspaces. We say that a collection of operators F on a Banach lattice X satisfies condition ( ) if there exists a closed ball B(x0,r) in X such that x0 > 0 and kx0k > r, and for every sequence (xn) in B(x0,r) (0,x0) there exists a subsequence (xni) and a sequence Ki 2 F such that Kixni converges to a non-zero vector. Let Q be a positive quasinilpotent operator on X, one-to-one, with dense range. Denote hQ) = {T > 0 : TQ 6 QT}. If either the set of all operators dominated by Q or the set of all contractions in hQ) satisfies ( ), then hQ) has a common invariant subspace. We also show that if Q is a one-to-one quasinilpotent interval preserving operator on C0(), then hQ) has a common invariant subspace. Lomonosov proved in (Lom73) that if T is not a multiple of the identity and com- mutes with a non-zero compact operator K, then T has a hyperinvariant subspace, that is, a proper closed nontrivial subspace invariant under every operator S in the commutant {T} 0 = {S 2 L(X) : ST = TS}. There has been numerous extensions and generalizations of the result of Lomonosov. In particular, Abramovich, Aliprantis, and Burkinshaw produced several generalizations of Lomonosov's theorem for Banach lattice setting (AAB93, AAB94, AAB98), see also (AA02). In these generalizations commutation relations are substituted by a super-commutation relation ST 6 TS or ST > TS and domination 0 6 K 6 T. They proved a series of results of the fol- lowing type: if S is related to a compact operator via a certain rather loose chain of super-commutations and dominations, then S has an invariant subspace. Ansari and Enflo (AE98) have recently introduced the so-called technique of minimal vectors in order to prove the existence of invariant subspaces for certain classes of operators on a Hilbert space. The method was later modified so that it could be used in arbitrary Banach spaces in (JKP03, And03, CPS04, Tr04). In particular, the method of minimal vectors allows to prove Lomonosov-type results where a compact operator is replaced with a family of operators that mimic a compact operator.

In this paper, we push forward the conjecture on a noncommutative version of the maximal ergodic theorem for positive contractions due to Ackoglu. That is, we establish the one‐sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative ‐spaces for a fixed , which particularly applies to positive isometries, the convex hull of positive Lamperti contractions, and also the power‐bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces . As an unexpected consequence, we deduce a completely bounded version of Ackoglu's theorem by making use of Ackoglu's original dilation. We also observe that the concrete examples of positive contractions without Akcoglu's dilation, which were constructed by Junge–Le Merdy [J. Funct. Anal. 249 (2007), no. 1, 220–252.], still satisfy the maximal ergodic inequalities. Together with the class of power‐bounded positive invertible operators considered by Hong–Liao–Wang [Duke Math. J. (2020). In press], we thereby verify the noncommutative Ackoglu ergodic theorem for all the positive operators that have naturally appeared in the literature up to the moment of writing. Based on the noncommutative Calderón transference principle established in [Duke Math. J. (2020). In press], the general pattern of the demonstration follows from the classical one due to Kan, but several new ideas are absolutely necessary in the noncommutative setting. Let us just mention three of them: the argument for the maximal ergodic theorem for positive isometries is highly nontrivial compared to the classical case; the novel simultaneous dilation property is indispensable to get the ergodic theorem for the convex hull of positive Lamperti contractions; numerous adjustments are truly needed in this new setting to conclude a desired structural theorem for positive doubly Lamperti operators since the orthogonal relations of operator algebras are completely different from those in classical measure theory.

Book review

We give some results about quotients of regular operators on Banach lattices by the linear span of the positive M-weakly and positive L-weakly compact operators. We also present a representation of the quotient space created by the linear span of the positive L-weakly compact operators.