A Lattice-Theoretic Description of the Lattice of Hyperinvariant Subspaces of a Linear Transformation

1976 ◽  
Vol 28 (5) ◽  
pp. 1062-1066 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Partial Order ◽  
Linear Transformation ◽  
Complete Lattice ◽  
Lattice Structure ◽  
Invariant Subspaces ◽  
Complex Hilbert Space ◽  
The Family ◽  
Hyperinvariant Subspaces ◽  
Theoretic Description

If A is a (linear) transformation acting on a (finitedimensional, non-zero, complex) Hilbert space H the family of (linear) subspaces of H which are invariant under A is denoted by Lat A. The family of subspaces of H which are invariant under every transformation commuting with A is denoted by Hyperlat A. Since A commutes with itself we have Hyperlat A ⊆ Lat A. Set-theoretic inclusion is an obvious partial order on both these families of subspaces. With this partial order each is a complete lattice; joins being (linear) spans and meets being set-theoretic intersections. Also, each has H as greatest element and the zero subspace (0) as least element. With this lattice structure being understood, Lat A (respectively Hyperlat A) is called the lattice of invariant (respectively, hyper invariant) subspaces of A.

scholarly journals Monomial codes seen as invariant subspaces

2017 ◽  
Vol 15 (1) ◽  
pp. 1099-1107 ◽  
Author(s):  
María Isabel García-Planas ◽  
Maria Dolors Magret ◽  
Laurence Emilie Um
Keyword(s):  
Finite Field ◽  
Linear Algebra ◽  
Linear Transformation ◽  
Cyclic Codes ◽  
Invariant Subspaces ◽  
Hyperinvariant Subspaces ◽  
Computationally Expensive ◽  
The Relationship

Abstract It is well known that cyclic codes are very useful because of their applications, since they are not computationally expensive and encoding can be easily implemented. The relationship between cyclic codes and invariant subspaces is also well known. In this paper a generalization of this relationship is presented between monomial codes over a finite field 𝔽 and hyperinvariant subspaces of 𝔽n under an appropriate linear transformation. Using techniques of Linear Algebra it is possible to deduce certain properties for this particular type of codes, generalizing known results on cyclic codes.

On Operator Algebras and Invariant Subspaces

1969 ◽  
Vol 21 ◽  
pp. 1178-1181 ◽  
Author(s):  
Chandler Davis ◽  
Heydar Radjavi ◽  
Peter Rosenthal
Keyword(s):  
Hilbert Space ◽  
Elementary Proof ◽  
Operator Algebras ◽  
Equivalent Form ◽  
Invariant Subspaces ◽  
Complex Hilbert Space ◽  
Closed Algebra ◽  
Weakly Closed ◽  
Special Case

If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies .Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.)ARVESON's THEOREM. Ifis a weakly closed algebra which contains an m.a.s.a.y and if Lat, then is the algebra of all operators on .A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.

scholarly journals Latent Complete-Lattice Structure of Hilbert-Space Projectors

Quanta ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 1
Author(s):  
Fedor Herbut
Keyword(s):  
Hilbert Space ◽  
Complete Lattice ◽  
Lattice Structure ◽  
Infinite Dimensional ◽  
Mechanical Formalism ◽  
Quantum Mechanical Formalism ◽  
The One ◽  
Infinite Sums ◽  
Adjoint Operators ◽  
Countably Infinite

To uncover the hidden complete-lattice structure of Hilbert-space projectors, which is not seen by the operator operations and relations (algebraically), resort is taken to the ranges of projectors (to subspaces—to geometry). Taking the range of a projector is completed into a bijection of all projectors onto all subspaces of any finite or countably infinite dimensional Hilbert space. As a second step, this basic bijection is upgraded into an isomorphism of partially ordered sets utilizing the sub-projector relation on the one hand, and the subspace relation on the other. As a third and final step, the basic bijection is further upgraded to isomorphism of complete lattices. The complete-lattice structure is derived for subspaces, then, using the basic bijection, it is transferred to the set of all projectors. Some consequences in the quantum-mechanical  formalism are examined with particular attention to the infinite sums appearing in spectral decompositions of discrete self-adjoint operators with infinite spectra.Quanta 2019; 8: 1–10.

Generalized Bloch Mappings in Complex Hilbert Space

1977 ◽  
Vol 29 (2) ◽  
pp. 299-306 ◽  
Author(s):  
Fletcher D. Wicker
Keyword(s):  
Hilbert Space ◽  
Analytic Functions ◽  
Unit Disc ◽  
Bloch Function ◽  
Complex Hilbert Space ◽  
Function Concept ◽  
Bounded Analytic Functions ◽  
Bloch Functions ◽  
Normal Functions ◽  
The Family

Anderson, Clunie and Pommerenke defined and studied the family of Bloch functions on the unit disc (see [1]). This family strictly contains the space of bounded analytic functions. However, all Bloch functions are normal and therefore enjoy the “nice” properties of normal functions. The importance of the Bloch function concept is the combination of their richness as a family and their “nice” behavior.

Remarks on Invariant Subspace Lattices

1969 ◽  
Vol 12 (5) ◽  
pp. 639-643 ◽  
Author(s):  
Peter Rosenthal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Invariant Subspace ◽  
Complete Lattice ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Subspace Lattice ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Subspace Lattices

If A is a bounded linear operator on an infinite-dimensional complex Hilbert space H, let lat A denote the collection of all subspaces of H that are invariant under A; i.e., all closed linear subspaces M such that x ∈ M implies (Ax) ∈ M. There is very little known about the question: which families F of subspaces are invariant subspace lattices in the sense that they satisfy F = lat A for some A? (See [5] for a summary of most of what is known in answer to this question.) Clearly, if F is an invariant subspace lattice, then {0} ∈ F, H ∈ F and F is closed under arbitrary intersections and spans. Thus, every invariant subspace lattice is a complete lattice.

scholarly journals Best Approximations by Increasing Invariant Subspaces of Self-Adjoint Operators

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1918
Author(s):  
Oleh Lopushansky ◽  
Renata Tłuczek-Piȩciak
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Adjoint Operator ◽  
Invariant Subspaces ◽  
Accurate Estimate ◽  
Complex Hilbert Space ◽  
Best Approximations ◽  
Adjoint Operators ◽  
Increasing Sequences

The paper describes approximations properties of monotonically increasing sequences of invariant subspaces of a self-adjoint operator, as well as their symmetric generalizations in a complex Hilbert space, generated by its positive powers. It is established that the operator keeps its spectrum over the dense union of these subspaces, equipped with quasi-norms, and that it is contractive. The main result is an inequality that provides an accurate estimate of errors for the best approximations in Hilbert spaces by these invariant subspaces.

scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

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