Nontrivial Invariant Subspace Research Articles

Criteria for Davies irreducibility of Markovian quantum dynamics

The dynamics of Markovian open quantum systems are described by Lindblad master equations, generating a quantum dynamical semigroup. An important concept for such systems is (Davies) irreducibility, i.e. the question whether there exist non-trivial invariant subspaces. Steady states of irreducible systems are unique and faithful, i.e. they have full rank. In the 1970s, Frigerio showed that a system is irreducible if the Lindblad operators span a self-adjoint set with trivial commutant. We discuss a more general and powerful algebraic criterion, showing that a system is irreducible if and only if the multiplicative algebra generated by the Lindblad operators La and the operator iH+∑aLa†La , involving the Hamiltonian H, is the entire operator space. Examples for two-level systems, show that a change of Hamiltonian terms as well as the addition or removal of dissipators can render a reducible system irreducible and vice versa. Examples for many-body systems show that a large class of spin chains can be rendered irreducible by dissipators on just one or two sites. Additionally, we discuss the decisive differences between (Davies) reducibility and Evans reducibility for quantum channels and dynamical semigroups which has lead to some confusion in the recent physics literature, especially, in the context of boundary-driven systems. We give a criterion for quantum reducibility in terms of associated classical Markov processes and, lastly, discuss the relation of the main result to the stabilization of pure states and argue that systems with local Lindblad operators cannot stabilize pure Fermi-sea states.

Nontrivial invariant subspaces of linear operator pencils

AbstractIn this paper, we introduce the spherical polar decomposition of the linear pencil of an ordered pair $\mathbf {T}=(T_{1},T_{2})$ and investigate nontrivial invariant subspaces between the generalized spherical Aluthge transform of the linear pencil of $\mathbf {T}$ and the linear pencil of the original pair $\mathbf {T}$ of bounded operators with dense ranges.

Square roots of complex symmetric operators

In this paper we study square roots of complex symmetric operators. In particular, we prove that if T∈L(H) is a square root of a complex symmetric operator, then T∗ has the single-valued extension property if and only if so does T. Moreover, in this case, T has the Bishop's property (β) if and only if T is decomposable. Finally, we show that if T has a nontrivial hyperinvariant subspace, then T∗ has a nontrivial invariant subspace.

Positive answer to the invariant and hyperinvariant subspaces problems for hyponormal operators

Abstract The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

Does a typical ℓ_{𝑝}-space contraction have a non-trivial invariant subspace?

Given a Polish topology τ \tau on B 1 ( X ) \mathcal {B}_{1}(X) , the set of all contraction operators on X = ℓ p X=\ell _p , 1 ≤ p > ∞ 1\le p>\infty or X = c 0 X=c_0 , we prove several results related to the following question: does a typical T ∈ B 1 ( X ) T\in \mathcal {B}_{1}(X) in the Baire Category sense has a non-trivial invariant subspace? In other words, is there a dense G δ G_\delta set G ⊆ ( B 1 ( X ) , τ ) \mathcal G\subseteq (\mathcal {B}_{1}(X),\tau ) such that every T ∈ G T\in \mathcal G has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the Strong ∗ ^* Operator Topology.

Subspace Decomposition for Graphon LQR: Applications to VLSNs of Harmonic Oscillators

Graphon control has been proposed and developed in [1]-[3] to approximately solve control problems for very large-scale networks (VLSNs) of linear dynamical systems based on graphon limits. This paper provides a solution method based on invariant subspace decompositions for a class of graphon linear quadratic regulation (LQR) problems where the local dynamics share homogeneous parameters but the graphon couplings may be heterogeneous among the coupled agents. Graphon couplings in this paper appear in states, controls and costs, and these couplings may be represented by different graphons. By exploring a common invariant subspace of the couplings, the original problem is decomposed into a network coupled LQR problem of finite dimension and a decoupled infinite dimensional LQR problem. A centralized optimal solution, and a nodal collaborative optimal control solution where each agent computes its part of the optimal solution locally, are established. The application of these solutions to finite network LQR problems may be via (i) the graphon control methodology [3], or (ii) the representation of finite LQR problems as special cases of graphon LQR problems. The complexity of these solutions involves solving one nd X nd dimensional Riccati equation and one n X n Riccati equation, where n is the dimension of each nodal agent state and d is the dimension of the nontrivial common invariant subspace of the coupling operators, whereas a direct approach involves solving an nN X nN dimensional Riccati equation, where N is the size of the network. For situations where the graphon couplings do not admit exact low-rank representations, approximate control is developed based on low-rank approximations. Finally, an application to the regulation of harmonic oscillators coupled over large networks with uncertainties is demonstrated.

Backward Shift and Nearly Invariant Subspaces of Fock-type Spaces

Abstract We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces ${\mathcal{F}}_W^p$, whose weight is not necessarily radial. We show that in the spaces ${\mathcal{F}}_W^p$, which contain the polynomials as a dense subspace (in particular, in the radial case), all nontrivial backward shift invariant subspaces are of the form $\mathcal{P}_n$, that is, finite-dimensional subspaces consisting of polynomials of degree at most $n$. In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to zero exponential type), we establish an analogue of de Branges’ ordering theorem. We then construct examples that show that the result fails for general Fock-type spaces of larger growth.

Quantitative and qualitative estimates on the norm of products of polynomials

When for the first time, in 1987, a Banach space X and a bounded operator T: X → X without nontrivial invariant subspaces was constructed, one of the many tools used was a series of estimates on the norm of a product of polynomials. Here, we continue this study of estimates on the norm of a product of polynomials by, on the one hand, extending some results due to Beauzamy and Enflo and, on the other, observing that an inequality by Borwein and Erdelyi holds in a more general context.

The invariant subspace problem for the space of smooth functions on the real line

We construct a continuous linear operator acting on the space of smooth functions on the real line without non-trivial invariant subspaces. This is a first example of such an operator acting on a Fréchet space without a continuous norm. The construction is based on the ideas due to C. Read who constructed a continuous operator without non-trivial invariant subspaces on the Banach space ℓ1.

Irreducibility of the monodromy representation of Lauricella's $F_C$

Let $E_C$ be the hypergeometric system of differential equations satisfied by Lauricella's hypergeometric series $F_C$ of $m$ variables. This system is irreducible in the sense of $D$-modules if and only if $2^{m+1}$ non-integral conditions for parameters are satisfied. We find a linear transformation of the classically known $2^m$ solutions so that the transformed ones always form a fundamental system of solutions under the irreducibility conditions. By using this fundamental system, we give an elementary proof of the irreducibility of the monodromy representation of $E_C$. When one of the conditions is not satisfied, we specify a non-trivial invariant subspace, which implies that the monodromy representation is reducible in this case.

On the local monodromy of $A$-hypergeometric functions and some monodromy invariant subspaces

We obtain an explicit formula for the characteristic polynomial of the local monodromy of $A$-hypergeometric functions with respect to small loops around a coordinate hyperplane $x\_i = 0$. This formula is similar to the one obtained by Ando, Esterov and Takeuchi for the local monodromy at infinity. Our proof is combinatorial and can be adapted to provide an alternative proof for the latter formula as well. On the other hand, we also prove that the solution space at a nonsingular point of certain irregular and irreducible $A$-hypergeometric $D$-modules has a nontrivial global monodromy invariant subspace.

Properties of absolute-*-k-paranormal operators and contractions for *-A(k) operators

First, we see if $T$ is absolute-$*$-$k$-paranormal for $k\geq 1$, then $T$ is a normaloid operator. We also see some properties of absolute-$*$-$k$-paranormal operator and $*$-$\mathcal{A}(k)$ operator. Then, we will prove the spectrum continuity of the class $*$-$\mathcal{A}(k)$ operator for $k>0$. Moreover, it is proved that if $T$ is a contraction of the class $*$-$\mathcal{A}(k)$ for $k>0$, then either $T$ has a nontrivial invariant subspace or $T$ is a proper contraction, and the nonnegative operator $$D=\left(T^{*}|T|^{2k}T\right)^{\frac{1}{k+1}}-|T^{*}|^{2}$$ is a strongly stable contraction. Finally if $T\in *$-$\mathcal{A}(k)$ is a contraction for $k>0,$ then $T$ is the direct sum of a unitary and $C_{\cdot 0}$ (c.n.u) contraction.

Taylor spectra and common invariant subspaces through the Duggal and generalized Aluthge transforms for commuting n-tuples of operators

n the first part of this paper, we introduce two notions of multivariable Duggal transforms (toral and spherical), and study their basic properties including hyponormality and norm-continuity. In the second part, we study how the Taylor spectrum and Taylor essential spectrum of 2-variable weighted shifts behave under the toral and spherical Duggal transforms including generalized Aluthge transforms. In the last part, we investigate nontrivial common invariant subspaces between the toral (respectively spherical) Duggal transform and the original n-tuple of bounded operators with dense ranges. We also study the sets of common invariant subspaces among them.

Invariant subspaces for non-normable Fréchet spaces

A Fréchet space X satisfies the Hereditary Invariant Subspace (resp. Subset) Property if for every closed infinite-dimensional subspace M in X, each continuous operator on M possesses a non-trivial invariant subspace (resp. subset). In this paper, we exhibit a family of non-normable separable infinite-dimensional Fréchet spaces satisfying the Hereditary Invariant Subspace Property and we show that many non-normable Fréchet spaces do not satisfy this property. We also state sufficient conditions for the existence of a continuous operator without non-trivial invariant subset and deduce among other examples that there exists a continuous operator without non-trivial invariant subset on the space of entire functions H(C).

Some invariant subspaces for w-hyponormal operators

ABSTRACTIn this paper, we prove that if is w-hyponormal, then the quasinilpotent part of T is given by for all sufficiently large integers p, where . We prove that if T is w-hyponormal and the spectrum is finite, then T is algebraic. In addition, we prove that if is w-hyponormal and has decomposition property , then T has a non-trivial invariant closed linear subspace. Also, we obtain that such an operator with rich spectrum has a nontrivial invariant subspace. Moreover, we consider invariant and hyperinvariant subspaces for w-hyponormal operators.

Triangular n-isometric operators

In this paper, we introduce the class of triangular n-isometric operators and study various properties. We show that every triangular n-isometric operator is subscalar of order 2n; in particular, every isometric operator is subscalar of order two. Consequently, if the spectrum of a triangular n-isometric operator T has a nonempty interior in , then T has a nontrivial invariant subspace. We also examine the hyperinvariant subspace problem for triangular n-isometric operators. Some spectral properties of this class of operators are also presented.

Wandering subspaces and quasi-wandering subspaces in the Hardy–Sobolev spaces

In this paper, we prove that for −1/2 ≤ β ≤ 0, suppose M is an invariant subspaces of the Hardy–Sobolev spaces H β 2(D) for T z β , then M ⊖ zM is a generating wandering subspace of M, that is, $$M = {\left[ {M \ominus zM} \right]_{T_z^\beta }}$$ . Moreover, any non-trivial invariant subspace M of H β 2(D) is also generated by the quasi-wandering subspace P M T z β M⊥, that is, $$M = {\left[ {{P_M}T_z^\beta {M^ \bot }} \right]_{T_z^\beta }}$$ .

Aluthge Transforms and Common Invariant Subspaces for a Commuting $$\varvec{n}$$ n -Tuple of Operators

It is well known that a bounded operator with dense range has a nontrivial invariant subspace if and only if its Aluthge transform does. Recently, R. Curto and Jasang Yoon have introduced the toral and spherical Aluthge transforms for commuting pairs (with particular emphasis on spherically quasinormal and spherically isometric 2-variable weighted shifts) and studied their basic properties. In this paper, we first investigate nontrivial common invariant subspaces between the toral (resp. spherical) Aluthge transform and the original n-tuple of bounded operators with dense ranges. We next study the sets of common invariant subspaces among them. Finally, we give an example which shows that the sets of common invariant subspaces between two Aluthge transforms (toral and spherical) are not isomorphic, even though they are equivalent with respect to the property of existence of a common nontrivial invariant subspace.

Triangularizability of trace-class operators with increasing spectrum

For any measurable set E of a measure space (X,μ), let PE be the (orthogonal) projection on the Hilbert space L2(X,μ) with the range ranPE={f∈L2(X,μ):f=0a.e. onEc} that is called a standard subspace of L2(X,μ). Let T be an operator on L2(X,μ) having increasing spectrum relative to standard compressions, that is, for any measurable sets E and F with E⊆F, the spectrum of the operator PET|ranPE is contained in the spectrum of the operator PFT|ranPF. In 2009, Marcoux, Mastnak and Radjavi asked whether the operator T has a non-trivial invariant standard subspace. They answered this question affirmatively when either the measure space (X,μ) is discrete or the operator T has finite rank. We study this problem in the case of trace-class kernel operators. We also slightly strengthen the above-mentioned result for finite-rank operators.

An invariant subspace problem for multilinear operators on Banach spaces and algebras

This paper is concerned with the study of invariant subspace problems for nonlinear operators on Banach spaces/algebras. Our study reveals that one faces unprecedented challenges such as lack of vector space structure and unbounded spectral sets when tackling invariant subspace problems for nonlinear operators via spectral information. To bypass some of these challenges, we modified an eigenvalue problem for nonlinear operators to cater for the structural properties of nonlinear operators and then established that nonlinear operators of finite type on a complex Banach algebra have nontrivial invariant subspaces.