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

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Salah Mecheri
Keyword(s):  
Invariant Subspace ◽  
Operator Theory ◽  
Affirmative Answer ◽  
Hyponormal Operator ◽  
Hyperinvariant Subspace ◽  
Large Classes ◽  
Infinite Dimensional ◽  
Hyponormal Operators ◽  
Scalar Operator ◽  
Hyperinvariant Subspaces

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.

scholarly journals Some remarks on the invariant subspace problem for hyponormal operators

2001 ◽  
Vol 28 (6) ◽  
pp. 359-365
Author(s):  
Vasile Lauric
Keyword(s):  
Invariant Subspace ◽  
Invariant Subspaces ◽  
Hyponormal Operator ◽  
Hyponormal Operators ◽  
Subspace Problem ◽  
Invariant Subspace Problem

We make some remarks concerning the invariant subspace problem for hyponormal operators. In particular, we bring together various hypotheses that must hold for a hyponormal operator without nontrivial invariant subspaces, and we discuss the existence of such operators.

scholarly journals A remark on the essential spectra of quasi-similar dominant contractions

1989 ◽  
Vol 31 (2) ◽  
pp. 165-168
Author(s):  
B. P. Duggal
Keyword(s):  
Complex Hilbert Space ◽  
Normal Extension ◽  
Inclusion Relation ◽  
Linear Transformations ◽  
Essential Spectra ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Hyponormal Operators ◽  
Image Position ◽  
Dominant Operator

We consider operators, i.e. bounded linear transformations, on an infinite dimensional separable complex Hilbert space H into itself. The operator A is said to be dominant if for each complex number λ there exists a number Mλ(≥l) such that ∥(A – λ)*x∥ ≤ Mλ∥A – λ)x∥ for each x∈H. If there exists a number M≥Mλ for all λ, then the dominant operator A is said to be M-hyponormal. The class of dominant (and JW-hyponormal) operators was introduced by J. G. Stampfli during the seventies, and has since been considered in a number of papers, amongst then [7], [11]. It is clear that a 1-hyponormal is hyponormal. The operator A*A is said to be quasi-normal if Acommutes with A*A, and we say that A is subnormal if A has a normal extension. It is known that the classes consisting of these operators satisfy the following strict inclusion relation:

On the Hyperinvariant Subspace Problem. IV

2008 ◽  
Vol 60 (4) ◽  
pp. 758-789 ◽  
Author(s):  
H. Bercovici ◽  
C. Foias ◽  
C. Pearcy
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Essential Spectrum ◽  
Complete Lattice ◽  
Unit Disc ◽  
Diagonal Operator ◽  
Hyperinvariant Subspace ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Subspace Problem

AbstractThis paper is a continuation of three recent articles concerning the structure of hyperinvariant subspace lattices of operators on a (separable, infinite dimensional) Hilbert space . We show herein, in particular, that there exists a “universal” fixed block-diagonal operator B on such that if ε > 0 is given and T is an arbitrary nonalgebraic operator on , then there exists a compact operator K of norm less than ε such that (i) Hlat(T) is isomorphic as a complete lattice to Hlat(B + K) and (ii) B + K is a quasidiagonal, C00, (BCP)-operator with spectrum and left essential spectrum the unit disc. In the last four sections of the paper, we investigate the possible structures of the hyperlattice of an arbitrary algebraic operator. Contrary to existing conjectures, Hlat(T) need not be generated by the ranges and kernels of the powers of T in the nilpotent case. In fact, this lattice can be infinite.

scholarly journals Hyperinvariant subspaces for some operators on locally convex spaces

2000 ◽  
Vol 23 (12) ◽  
pp. 807-814
Author(s):  
Edvard Kramar
Keyword(s):  
Locally Convex Spaces ◽  
Bounded Operators ◽  
Hyperinvariant Subspace ◽  
Hyperinvariant Subspaces ◽  
Locally Convex ◽  
Convex Spaces

Some results concerning hyperinvariant subspaces of some operators on locally convex spaces are considered. Denseness of a class of operators which have a hyperinvariant subspace in the algebra of locally bounded operators is proved.

scholarly journals Invariant subspace lattices of Lambert's weighted shifts

1982 ◽  
Vol 33 (1) ◽  
pp. 135-142
Author(s):  
B. S. Yadav ◽  
S. Chatterjee
Keyword(s):  
Hilbert Space ◽  
Invariant Subspace ◽  
Bounded Sequence ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weighted Shifts ◽  
Positive Real ◽  
Subspace Lattice ◽  
Bounded Linear ◽  
Infinite Dimensional

AbstractLet B(H) be the Banach algebra of all (bounded linear) operators on an infinite-dimensional separable complex Hilbert space H and let be a bounded sequence of positive real numbers. For a given injective operator A in B(H) and a non-zero vector f in H, we put We define a weighted shift Tw with the weight sequence on the Hilbert space 12 of all square-summable complex sequences by . The main object of this paper is to characterize the invariant subspace lattice of Tw under various nice conditions on the operator A and the sequence .

scholarly journals The Invariant Subspace Problem for Non-Archimedean Banach Spaces

2008 ◽  
Vol 51 (4) ◽  
pp. 604-617 ◽  
Author(s):  
Wiesław Śliwa
Keyword(s):  
Banach Space ◽  
Invariant Subspace ◽  
Banach Spaces ◽  
Continuous Operator ◽  
Linear Continuous Operator ◽  
Infinite Dimensional ◽  
Countable Type ◽  
Closed Invariant Subspace ◽  
Subspace Problem ◽  
Invariant Subspace Problem

AbstractIt is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A. C. M. van Rooij and W. H. Schikhof in 1992.

Lomonosov's hyperinvariant subspace theorem for real spaces

1981 ◽  
Vol 89 (1) ◽  
pp. 129-133 ◽  
Author(s):  
N. D. Hooker
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Linear Operators ◽  
Continuous Linear ◽  
Hyperinvariant Subspace ◽  
Subspace Theorem ◽  
Hyperinvariant Subspaces ◽  
A New Technique ◽  
Continuous Linear Operators

In 1973, V.I.Lomonosov introduced a new technique for finding invariant and hyperinvariant subspaces for certain classes of (continuous, linear) operators on complex Banach spaces. Recall that a closed subspace M of the Banach space X is called hyperinvariant for the operator T if S(M) ⊂ M for every operator S which commutes with T.

scholarly journals HYPONORMAL OPERATORS, WEIGHTED SHIFTS AND WEAK FORMS OF SUPERCYCLICITY

2006 ◽  
Vol 49 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Frédéric Bayart ◽  
Etienne Matheron
Keyword(s):  
Banach Space ◽  
Unitary Operator ◽  
Dimensional Subspace ◽  
The Other ◽  
Hyponormal Operator ◽  
Weighted Shifts ◽  
One Dimensional ◽  
Other Hand ◽  
Hyponormal Operators

AbstractAn operator $T$ on a Banach space $X$ is said to be weakly supercyclic (respectively $N$-supercyclic) if there exists a one-dimensional (respectively $N$-dimensional) subspace of $X$ whose orbit under $T$ is weakly dense (respectively norm dense) in $X$. We show that a weakly supercyclic hyponormal operator is necessarily a multiple of a unitary operator, and we give an example of a weakly supercyclic unitary operator. On the other hand, we show that hyponormal operators are never $N$-supercyclic. Finally, we characterize $N$-supercyclic weighted shifts.

scholarly journals On the spectrum of n-tuples of p-hyponormal operators

1998 ◽  
Vol 40 (1) ◽  
pp. 123-131 ◽  
Author(s):  
B. P. Duggal
Keyword(s):  
Spectral Properties ◽  
Polar Decomposition ◽  
Partial Isometry ◽  
Hyponormal Operator ◽  
Square Root ◽  
Linear Transformations ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Algebra Of Operators ◽  
Number Of Authors

Let B(H) denote the algebra of operators (i.e., bounded linear transformations) on the Hilbert space H. A ∈ B (H) is said to be p-hyponormal (0<p<l), if (AA*)γ < (A*A)p. (Of course, a l-hyponormal operator is hyponormal.) The p-hyponormal property is monotonic decreasing in p and a p-hyponormal operator is q-hyponormal operator for all 0<q <p. Let A have the polar decomposition A = U |A|, where U is a partial isometry and |A| denotes the (unique) positive square root of A*A.If A has equal defect and nullity, then the partial isometry U may be taken to be unitary. Let ℋU(p) denote the class of p -hyponormal operators for which U in A = U |A| is unitary. ℋU(l/2) operators were introduced by Xia and ℋU(p) operators for a general 0<p<1 were first considered by Aluthge (see [1,14]); ℋU(p) operators have since been considered by a number of authors (see [3, 4, 5, 9, 10] and the references cited in these papers). Generally speaking, ℋU(p) operators have spectral properties similar to those of hyponormal operators. Indeed, let A ε ℋU(p), (0<p <l/2), have the polar decomposition A = U|A|, and define the ℋW(p + 1/2) operator  by A = |A|1/2U |A|l/2 Let  = V |Â| Â= |Â|1/2VÂ|ÂAcirc;|1/2. Then we have the following result.

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