Lattice Of Invariant Subspaces Research Articles

A note on a Halmos problem

AbstractWe address the existence of non‐trivial closed invariant subspaces of operators on Banach spaces whenever their square have or, more generally, whether there exists a polynomial with such that the lattice of invariant subspaces of is non‐trivial. In the Hilbert space setting, the ‐problem was posed by Halmos in the seventies and in 2007, Foias, Jung, Ko and Pearcy conjectured it could be equivalent to the Invariant Subspace Problem.

Idempotent vector spaces and their linear transformations

Abstract This article extends topics about linear algebra and operator theoretic linear transformations on complex vector spaces to those on bicomplex spaces. For example, Definition 3 for the first time defines algebraically idempotent vector spaces, which generalizes the standard definition of a vector space and which includes bicomplex vector spaces as a special case, along with its dimension and its basis in terms of a corresponding vectorial idempotent representation. The article also shows how an n × n n\times n bicomplex matrix’s idempotent representation leads to a bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, in a new way, the article rigorously defines “bicomplex Banach and Hilbert” spaces, and then it expands, for the first time, the theory of compact operators on complex Banach spaces to those on bicomplex Banach spaces. In these ways, the article indicates that the idempotent representation extends complex linear algebra and operator theory in a surprisingly generalized and straightforward way to vector space results with bicomplex and multicomplex scalars.

Nonlinear preservers problem of complex symmetric operators

Given a conjugation [Formula: see text] on a separable complex Hilbert space [Formula: see text], a bounded linear operator [Formula: see text] on [Formula: see text] is said to be [Formula: see text]-symmetric if [Formula: see text]. In this paper, we study some nonlinear preserving problems concerning [Formula: see text]-symmetric operators and diagonal operators, as a result, we also describe those nonlinear preservers of the lattice of invariant subspaces.

Invariant Synchrony Subspaces of Sets of Matrices

A synchrony subspace of $\mathbb{R}^{n}$ is defined by setting certain components of the vectors equal according to an equivalence relation. Synchrony subspaces invariant under a given set of square matrices ordered by inclusion form a lattice. Applications of these invariant synchrony subspaces include equitable and almost equitable partitions of the vertices of a graph used in many areas of graph theory, balanced and exo-balanced partitions of coupled cell networks, and coset partitions of Cayley graphs. We study the basic properties of invariant synchrony subspaces and provide many examples of the applications. We also present what we call the split and cir algorithm for finding the lattice of invariant synchrony subspaces. Our theory and algorithm is further generalized for nonsquare matrices. This leads to the notion of tactical decompositions studied for its application in design theory.

Constructing invariant subspaces as kernels of commuting matrices

Given an n×n matrix A over C and an invariant subspace N, a straightforward formula constructs an n×n matrix N that commutes with A and has N=ker⁡N. For Q a matrix putting A into Jordan canonical form, J=Q−1AQ, we get N=Q−1M where M= ker(M) is an invariant subspace for J with M commuting with J. In the formula M=PZT−1Pt, the matrices Z and T are m×m and P is an n×m row selection matrix. If N is a marked subspace, m=n and Z is an n×n block diagonal matrix, and if N is not a marked subspace, then m>n and Z is an m×m near-diagonal block matrix. Strikingly, each block of Z is a monomial of a finite-dimensional backward shift. Each possible form of Z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice L(A) for A.

Norms of vector functionals

We examine the question of when, and how, the norm of a vector functional on an operator algebra can be controlled by the invariant subspace lattice of the algebra. We introduce a related operator algebraic property, and show that it is satisfied by all von Neumann algebras and by all CSL algebras. We exhibit examples of operator algebras that do not satisfy the property or any scaled version of it.

The lattice of characteristic subspaces of an endomorphism with Jordan–Chevalley decomposition

Given an endomorphism A over a finite dimensional vector space having Jordan–Chevalley decomposition, the lattices of invariant and hyperinvariant subspaces of A can be obtained from the nilpotent part of this decomposition. We extend this result for lattices of characteristic subspaces. We also obtain a generalization of Shoda's theorem about the characterization of the existence of characteristic non hyperinvariant subspaces.

The invariant subspaces of the shift plus integer multiple of the Volterra operator on Hardy spaces

Cuckovic and Paudyal recently characterized the lattice of invariant subspaces of the shift plus a complex Volterra operator on the Hilbert space \(H^2\) on the unit disk. Motivated by the idea of Ong, in this paper, we give a complete characterization of the lattice of invariant subspaces of the shift operator plus a positive integer multiple of the Volterra operator on Hardy spaces \(H^p\), which essentially extends their works to the more general cases when \(1\le p<\infty \).

The lattices of invariant subspaces of a class of operators on the Hardy space

In the authors’ first paper, a Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. The current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator.

Volterra type operators on [formula omitted] spaces

Čučković and Paudyal characterized the lattice of invariant subspaces of the operator T in the Hardy–Hilbert space H2(D), where they studied the special case (when p=2) of the space Sp(D). We generalize some of their works to the general case when 1≤p<∞ and determine that M is an invariant subspace of T on Hp(D) if and only if Tz(M) is an invariant subspace of Mz on S0p(D), if and only if Tz(M) is a closed ideal of S0p(D). Furthermore, we provide certain Beurling-type invariant subspaces of Mz on Sp(D) and S0p(D). Then, we investigate the boundedness of the operators Tg and Ig on Sp(D). Finally, we investigate the spectrum of multiplication operator Mg on Sp(D), the isometric multiplication operators and the isometric zero-divisors on Sp(D).

A Note on Some Equivalences of Operators and Topology of Invariant Subspaces

In this paper we investigate the invariant and hyperinvariant subspace lattices of some operators. We give a lattice-theoretic description of the lattice of hyperinvariant subspaces of an operator in terms of its lattice of invariant subspaces. We also study the structure of these lattices for operators in certain equivalence classes of some equivalence relations.

Compact Composition Operators and Deddens Algebras

We consider the Deddens algebras associated to compact composition operators on the Hardy space \(H^2\) on the unit disk. When the compact composition operator corresponds to a function \(\varphi \) that satisfies \(\varphi (0)=0\) and \(\varphi '(0)\ne 0\), we show that the lattice of invariant subspaces of this algebra is \(\{0\}\cup \{z^n H^2: n=0,1,2,\ldots \}\). As a consequence, for this class of operators the associated Deddens algebra is weakly dense in the algebra of lower triangular matrices.

The operator algebra generated by the translation, dilation and multiplication semigroups

The weak operator topology closed operator algebra on L 2 ( R ) generated by the one-parameter semigroups for translation, dilation and multiplication by e i λ x , λ ≥ 0 , is shown to be a reflexive operator algebra, in the sense of Halmos, with invariant subspace lattice equal to a binest. This triple semigroup algebra, A p h , is antisymmetric in the sense that A ph ∩ A p h ⁎ = C I , it has a nonzero proper weakly closed ideal generated by the finite-rank operators, and its unitary automorphism group is R . Furthermore, the 8 choices of semigroup triples provide 2 unitary equivalence classes of operator algebras, with A ph and A p h ⁎ being chiral representatives.

Invariant subspaces of composition operators

The invariant subspace lattices of composition operators acting on H 2 , the Hilbert-Hardy space over the unit disc, are characterized in select cases. The lattice of all spaces left invariant by both a composition operator and the unilateral shift Mz (the multiplication operator induced by the co- ordinate function), is shown to be nontrivial and is completely described in particular cases. Given an analytic selfmap j of the unit disc, we prove that j has an angular derivative at some point on the unit circle if and only if Cj, the composition operator induced by j, maps certain subspaces in the invariant subspace lattice of Mz into other such spaces. A similar characterization of the existence of angular derivatives of j, this time in terms of Aj, the Aleksandrov operator induced by j, is obtained.

Reflexive Index of a Family of Sets

As a further study on re∞exive families of subsets, we introduce the re∞exive index for a family of subsets of a given set and show that the index of a flnite family of subsets of a flnite or countably inflnite set is always flnite. The re∞exive indices of some special families are also considered. Given a set X, let Sub(X) denote the set of all subsets of X and End(X) denote the set of all endomappings f : X i! X. For any A µ Sub(X) and F µ End(X) deflne Alg(A) = ff 2 End(X) : f(A) µ A for all A 2 Ag; Lat(F) = fA 2 Sub(X) : f(A) µ A for all f 2 Fg: A family A µ Sub(X) is called re∞exive if A = Lat(Alg(A)), or equivalently, A = Lat(F) for some F µ End(X). As was shown in (9), A µ Sub(X) is re∞exive ifi it is closed under arbitrary unions and intersections and contains the empty set and X. The re∞exive families F µ End(X) were also introduced and characterized as those subsemigroups L of (End(X);-) such that L is a lower set and contains all existing suprema of subsets of L with respect to a naturally deflned partial order on End(X). The similar work in functional analysis is on the re∞exive invariant subspace lattices and re∞exive operator algebras (1-6). For any A µ Sub(X), let ^ A = Lat(Alg(A)). Then ^ A is the smallest family of subsets containing A which is closed under arbitrary unions and intersections containing empty set ; and X, and ^

Reflexive operator algebras on Banach spaces

In this paper we study the reflexivity of a unital strongly closed algebra of operators with complemented invariant subspace lattice on a Banach space. We prove that if such an algebra contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property, then it is reflexive, i.e., it contains every operator that leaves invariant every closed subspace in the invariant subspace lattice of the algebra. In particular, such algebras coincide with their bicommutant. Let A B.X/ denote a strongly closed algebra of operators on the Banach space X. Suppose that A has the property that each of its invariant subspaces has an invariant complement. If A contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property as defined below, we prove that A is reflexive in the sense that it contains all the operators which leave its closed invariant subspaces invariant (Theorem 15). In particular such an algebra is equal to its bicommutant A 00 (Corollary 22). The problem of whether a strongly closed algebra of operators with complemented invariant subspace lattice is reflexive started to be studied in the sixties. This problem is a generalization of the invariant subspace problem in operator theory. Arveson [1967] introduced a technique for studying the particular case of transitive algebras on Hilbert spaces, namely the strongly closed algebras of operators on Hilbert spaces that have no nontrivial closed invariant subspaces. He proved that every transitive algebra that contains a maximal abelian von Neumann algebra coincides with the full algebra B.X/ if X is a complex Hilbert space. Douglas and Pearcy [1972] extended the result of Arveson to the case of transitive operator algebras containing an abelian von Neumann algebra of finite multiplicity. Hoover [1973] extended the result of Douglas and Pearcy to the case of reductive operator algebras on Hilbert spaces that contain C. Peligrad and M. Peligrad were supported in part by a Charles Phelps Taft Memorial Fund grant. M. Peligrad was also supported by NSF grant DMS-1208237. MSC2010: primary 47A15, 47B48; secondary 47C05.

Structural matrix algebras related to finite distributive lattices

In this paper, we investigate the relation between a structural matrix algebra and the lattice properties of its lattice of invariant subspaces, and reprove known results in a fresh and an explanatory way. Moreover, we also prove the theorem which is partially converse of Proposition 2.6 of [15].

Invariant subspaces of parabolic self-maps in the Dirichlet space

In this note, it is provided a complete description of the lattice of invariant subspaces of a composition operator acting on the Dirichlet space and whose inducing symbol is a parabolic non-automorphism.

On the Topology of Invariant Subspaces of a Shift of Higher Multiplicity

Following Beurling’s theorem R. Douglas and C. Pearcy have studied topology of invariant subspace lattice. R.Yang offered a description of path connected components of invariant subspace lattice for shift of multiplicity one. This paper generalizes the result to arbitrary finite multiplicity. We show that there exists a one to one correspondence between the invariant subspace lattice of shift of arbitrary finite multiplicity and the space of inner functions.

Complemented invariant subspaces of structural matrix algebras

In this paper, we explore when the lattice of invariant subspaces of a structural matrix algebra can be complemented. We give several equivalent conditions for this lattice to be a Boolean algebra.