Nontrivial Invariant Subspace Research Articles

The invariant subspace structure of $L^2(\Bbb T^2)$ for certain von Neumann algebras

In this note, we study invariant subspaces of $L^2(\Bbb T^2)$ with respect to certain von Neumann algebras. We give a characterization of Beurling-type left-invariant subspaces of $L^2(\Bbb T^2)$. We also give a structure theorem of a non-trivial two-sided invariant subspace of $L^2(\Bbb T^2)$.

Joint local quasinilpotence and common invariant subspaces

In this article we obtain some positive results about the existence of a common nontrivial invariant subspace forN-tuples of not necessarily commuting operators on Banach spaces with a Schauder basis. The concept of joint quasinilpotence plays a basic role. Our results complement recent work by Kosiek [6] and Ptak [8].

Characteristic functions and joint invariant subspaces

Let T : = [ T 1 , … , T n ] be an n-tuple of operators on a Hilbert space such that T is a completely non-coisometric row contraction. We establish the existence of a “one-to-one” correspondence between the joint invariant subspaces under T 1 , … , T n , and the regular factorizations of the characteristic function Θ T associated with T. In particular, we prove that there is a non-trivial joint invariant subspace under the operators T 1 , … , T n , if and only if there is a non-trivial regular factorization of Θ T . We also provide a functional model for the joint invariant subspaces in terms of the regular factorizations of the characteristic function, and prove the existence of joint invariant subspaces for certain classes of n-tuples of operators. We obtain criteria for joint similarity of n-tuples of operators to Cuntz row isometries. In particular, we prove that a completely non-coisometric row contraction T is jointly similar to a Cuntz row isometry if and only if the characteristic function of T is an invertible multi-analytic operator.

Invariant subspaces of X** under the action of biconjugates

We study conditions on an infinite dimensional separable Banach space X implying that X is the only non-trivial invariant subspace of X** under the action of the algebra \(\mathbb{A}(X)\) of biconjugates of bounded operators on \(X: \mathbb{A}(X) = \{ T^{**} :T \in \mathcal{B}(X)\} \). Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of c 0, and show in particular that any space which does not contain ℓ1 and has property (u) of Pelczynski is simple.

Semitransitive subspaces of matrices

A set of matrices S⊆ Mn(F) is said to be semitransitive if for any two nonzero vectors x,y ∈ F n ,t here exi sts am atri xA ∈S such that either Ax = y or Ay = x. In this paper various properties of semitransitive linear subspaces of Mn(F) are studied. In particular, it is shown that every semitransitive subspace of matrices has a cyclic vector. Moreover, if |F |≥ n, it always contains an invertible matrix. It is proved that there are minimal semitransitive matrix spaces without any nontrivial invariant subspace. The structure of minimal semitransitive spaces and triangularizable semitransitive spaces is also studied. Among other results it is shown that every triangularizable semitransitive subspace contains a nonzero nilpotent.

On the spectral multiplicity of a direct sum of operators

a nonnegative integer or ∞. Clearly, A is cyclic if and only if μ(A) = 1. The spectral multiplicity is an important invariant of operators, and it plays a key role in operator theory and its applications. Clearly, the notion of cyclic subspace is important in connection with the general problem of existence of a nontrivial invariant subspace, because an operator A ∈ L(X) has no nontrivial invariant subspace if and only if x ∈ Cyc(A) for every x ∈ X\{0}. Cyclic vectors are important in weighted polynomial approximation theory. (More details can be found in [6].) In this article we calculate the spectral multiplicity of the direct sum T ⊕A, where T is a weighted shift operator on a Banach space Y continuously embedded in lp, and A a suitable bounded operator on a Banach space X (Section 2). Note that the main result of Section 2, Theorem 1, generalizes and strengthens some results of the author in [3, Theorem], [4, Theorem 3] and [5, Theorem 3]. First we introduce some notations and definitions. If {ei}i≥0 is a sequence of vectors in a Banach space X, we say that {ei}i≥0 is uniformly minimal if there exists a constant δ > 0 such that

Sub-n-normal Operators

In this paper we introduce the class of sub-n-normal operators. By definition, such an operator is the restriction to an invariant subspace of an n-normal operator, and thus the sub-n-normal operators form a larger class than the subnormal operators. We obtain some modest structure theorems and contrast sub-n-normal operators with sub-Jordan operators. Finally we show that a sub-n-normal operator with rich spectrum has a nontrivial invariant subspace.

CONTRACTIONS OF CLASS Q AND INVARIANT SUBSPACES

A Hilbert Space operator T is of class Q if <TEX>$T^2{\ast}T^2-2T{\ast}T + I$</TEX> is nonnegative. Every paranormal operator is of class Q, but class-Q operators are not necessarily normaloid. It is shown that if a class-Q contraction T has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator Q = <TEX>$T^2{\ast}T^2-2T{\ast}T + I$</TEX> also is a proper contraction.

Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action

It is proved that a Lie algebra of compact operators with a non-zero Volterra ideal is reducible (has a nontrivial invariant subspace). A number of other criteria of reducibility for collections of operators is obtained. The results are applied to the structure theory of Lie algebras of compact operators and normed Lie algebras with compact adjoint action.

Beyond hyperinvariance for compact operators

We introduce a new class of operator algebras on Hilbert space. To each bounded linear operator a spectral algebra is associated. These algebras are quite substantial, each containing the commutant of the associated operator, frequently as a proper subalgebra. We establish several sufficient conditions for a spectral algebra to have a nontrivial invariant subspace. When the associated operator is compact this leads to a generalization of Lomonosov's theorem.

Contractions Satisfying the Absolute Value Property $$ |A|^2 \leq |A^2| $$

Let B(H) denote the algebra of operators on a complex Hilbert space H, and let U denote the class of operators \( A \in B(H) \) which satisfy the absolute value condition \( |A|^2 \leq |A^2| \). It is proved that if \( A \in \mathcal{U} \) is a contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the nonnegative operator \( D = |A^2| - |A|^2 \) is strongly stable. A Putnam-Fuglede type commutativity theorem is proved for contractions A in \( \mathcal{U} \), and it is shown that if normal subspaces of \( |A|^2 \leq |A^2| \). It is proved that if \( A \in \mathcal{U} \) are reducing, then every compact operator in the intersection of the weak closure of the range of the derivation \( \delta_{A}(X) = AX - XA \) with the commutant of A* is quasinilpotent.

The simplest proof of Burnside's theorem on matrix algebras

A very simple, short and self-contained proof is presented of Burnside's Theorem that every proper algebra of matrices over an algebraically closed field has a non-trivial invariant subspace.

Invariant subspaces for polynomially bounded operators

Let T be a polynomially bounded operator on a Banach space X whose spectrum contains the unit circle. Then T ∗ has a nontrivial invariant subspace. In particular, if X is reflexive, then T itself has a nontrivial invariant subspace. This generalizes the well-known result of Brown, Chevreau, and Pearcy for Hilbert space contractions.

Square roots of (BCP)-operators

In this note we consider a class of square roots of invertible (BCP)-operators whose structure is pertinent to the question whether every operator on Hilbert space whose norm and spectral radius coincide has a nontrivial invariant subspace.

On common invariant subspaces for commuting contractions with rich spectrum

In this paper we show that an N-tuple of commuting contractions T = (T 1 ,..., T N ) acting on a separable, complex Hilbert space H having the polydisk (D N ) as a spectral set and dominating Harte spectrum, has a nontrivial common invariant subspace (i.e., a proper subspace {0} ¬= M ⊂ H, such that T j M ⊂ M for j = 1,..., N). We do not need to assume any kind of C 0 . condition for any member of our N-tuple.

PARANORMAL CONTRACTIONS AND INVARIANT SUBSPACES

It is shown that if a paranormal contraction T has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator Q = T/sup 2*/T/sup 2/ - 2T/sup */T ＋ I also is a proper contraction. If a quasihyponormal contraction has no nontrivial invariant subspace then, in addition, its defect operator D is a proper contraction and its itself-commutator is a trace-class strict contraction. Furthermore, if one of Q or D is compact, then so is the other, and Q and D are strict ontraction.

PROPERTIES OF A κTH ROOT OF A HYPONORMAL OPERATOR

In this paper, we study some properties of (equation omitted) (defined below). In particular we show that an operator T <TEX>$\in$</TEX>(equation omitted) satisfying the translation invariant property is hyponormal and an invertible operator T <TEX>$\in$</TEX> (equation omitted) and its inverse T<TEX>$^{-1}$</TEX> have a common nontrivial invariant closed set. Also we study some cases which have nontrivial invariant subspaces for an operator in (equation omitted).

The simplest proof of Burnside's theorem on matrix algebras

A very simple, short and self-contained proof is presented of Burnside's Theorem that every proper algebra of matrices over an algebraically closed field has a non-trivial invariant subspace.

Square roots of semihyponormal operators have scalar extensions

In this paper, we study some properties of ( SH ) , i.e., square roots of semihyponormal operators. In particular we show that an operator T∈( SH ) has a scalar extension, i.e., is similar to the restriction to an invariant subspace of a (generalized) scalar operator (in the sense of Colojoară–Foiaş). As a corollary, we obtain that an operator T∈( SH ) has a nontrivial invariant subspace if its spectrum has interior in the plane.

Aluthge transformations and invariant subspaces of p-hyponormal operators

It is unknown at present whether every hyponormal operator has a nontrivial invariant subspace. Many authors presented conditions for a hyponormal operator to have nontrivial invariant subspaces. In this paper, we give a p-hyponormal version of Nakamura's result [7] by using the principal functions.