scholarly journals On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Positive Definite ◽  
Finite Dimensional ◽  
Positive Semidefiniteness ◽  
Order Isomorphisms ◽  
Positive Definite Operators ◽  
Finite Dimensional Case ◽  
The One ◽  
Adjoint Operators ◽  
Similar Assertion ◽  
Usual Order

We prove that there is no bijective map between the set of all positive definite operators and the set of all self-adjoint operators on a Hilbert space with dimension greater than 1 which preserves the usual order (the one coming from the concept of positive semidefiniteness) in both directions. We conjecture that a similar assertion is true for general noncommutativeC*-algebras and present a proof in the finite dimensional case.

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

scholarly journals Complementary Lagrangians in infinite dimensional symplectic Hilbert spaces

2005 ◽  
Vol 77 (4) ◽  
pp. 589-594 ◽  
Author(s):  
Paolo Piccione ◽  
Daniel V. Tausk
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Countable Family ◽  
Spectral Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Lagrangian Subspaces ◽  
Finite Dimensional Case ◽  
Adjoint Operators

We prove that any countable family of Lagrangian subspaces of a symplectic Hilbert space admits a common complementary Lagrangian. The proof of this puzzling result, which is not totally elementary also in the finite dimensional case, is obtained as an application of the spectral theorem for unbounded self-adjoint operators.

scholarly journals Skew compressions of positive definite operators and matrices

2020 ◽  
Vol 36 (36) ◽  
pp. 400-410
Author(s):  
Matteo Polettini ◽  
Albrecht Böttcher
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Singular Values ◽  
Positive Definite ◽  
Graph Theoretic ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Positive Definite Operators ◽  
Finite Dimensional Hilbert Space

The paper is devoted to results connecting the eigenvalues and singular values of operators composed by $P^\ast G P$ with those composed in the same way by $QG^{−1}Q^\ast$. Here $P +Q = I$ are skew complementary projections on a finite-dimensional Hilbert space and $G$ is a positive definite linear operator on this space. Also discussed are graph theoretic interpretations of one of the results.

scholarly journals Real parts of quasi-nilpotent operators

1979 ◽  
Vol 22 (3) ◽  
pp. 263-269 ◽  
Author(s):  
P. A. Fillmore ◽  
C. K. Fong ◽  
A. R. Sourour
Keyword(s):  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Dimensional Case ◽  
Nilpotent Operator ◽  
The Real ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Nilpotent Operators ◽  
Finite Dimensional Case ◽  
Adjoint Operators

The purpose of this paper is to answer the question: which self-adjoint operators on a separable Hilbert space are the real parts of quasi-nilpotent operators? In the finite-dimensional case the answer is: self-adjoint operators with trace zero. In the infinite dimensional case, we show that a self-adjoint operator is the real part of a quasi-nilpotent operator if and only if the convex hull of its essential spectrum contains zero. We begin by considering the finite dimensional case.

scholarly journals On Differences of Unitarily Equivalent Self-Adjoint Operators

1960 ◽  
Vol 4 (3) ◽  
pp. 103-107 ◽  
Author(s):  
C. R. Putnam
Keyword(s):  
Continuous Operator ◽  
Bounded Operators ◽  
Constant Multiple ◽  
Additional Hypothesis ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case ◽  
The Difference ◽  
Image Position ◽  
Adjoint Operators

1. All operators considered in this paper are bounded operators on a Hilbert space. In case A and B are self-adjoint, certain conditions on A, B and their differenceassuring the unitary equivalence of Aand B,have recently been obtained by Rosenblum [6] and Kato [2]. The present paper will consider the problem of investigating consequences of an assumed relation of type (2) for some unitary U together with an additional hypothesis that the difference H of (1) be non-negative, so thatFirst, it is easy to see that if only (2) and (3) are assumed, thereby allowing H = 0, relation (2) can hold for A arbitrary with U = I (identity) and B = A. If H = 0 in (3) is not allowed, however (an impossible assumption in the finite dimensional case, incidentally, since then the trace of H is zero and hence H = 0), it will be shown, among other things, that any unitary operator U for which (2) and (3) hold must have a spectrum with a positive measure (as a consequence of (i) of Theorem 2 below). Moreover A (hence B) cannot differ from a completely continuous operator by a constant multiple of the identity (Theorem 1). In case 0 is not in the point spectrum of H, then U is even absolutely continuous (see (iv) of Theorem 2). In § 4, applications to semi-normal operators will be given.

Spectral properties of two parameter eigenvalue problems II

1987 ◽  
Vol 106 (1-2) ◽  
pp. 39-51 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne
Keyword(s):  
Hilbert Space ◽  
Spectral Properties ◽  
Eigenvalue Problems ◽  
Separable Hilbert Space ◽  
Asymptotic Properties ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Two Parameter

SynopsisWe consider eigenvalues λ =(λ1, λ2) ∈R2 for the problem W(λ)x = 0, x ≠ 0, x ∈ H, where W(λ) = R + λ1V1 + λ2V2), and R, V1, V2 are self-adjoint operators on a separable Hilbert space H, R being bounded below with compact resolvent and V1, V2 being bounded. The i-th eigencurve Z1 is the set of eigenvalues λ, for which the i-th eigenvalue (counted according to multiplicity and in increasing order) of W(λ) vanishes. We study monotonic and asymptotic properties of Zi, and we give formulae for any asymptotes that exist. Additional results are given in the finite dimensional case.

Refined positivity theorem for semigroups generated by perturbed differential operators of second order with an application to Markov branching processes

1978 ◽  
Vol 83 (2) ◽  
pp. 253-259 ◽  
Author(s):  
H. Hering
Keyword(s):  
Branching Process ◽  
Differential Operators ◽  
Branching Processes ◽  
Bounded Operators ◽  
Underlying Space ◽  
Finite Dimensional ◽  
Limiting Behaviour ◽  
Finite Dimensional Case ◽  
The One ◽  
Branching Diffusions

The search for a result such as the one presented in this note was motivated by an application in the theory of Markov branching processes. The limiting behaviour of a Markov branching process is determined mainly by properties of the set of its first moments, usually given as a semigroup of non-negative, linear-bounded operators on a Banach space. The principal case is that in which these operators are in some sense primitive. If the underlying space is finite-dimensional, the case of primitivity is described to complete satisfaction by Perron's theorem. For more general spaces we have the well known extension of Perron's result by Kreĭn and Rutman (8). Unfortunately, the use of this extension in the theory of general branching processes has so far not led to limit theorems as strong as the best results known in the finite-dimensional case. At least for this specific purpose the classical Kreĭn-Rutman theorem seems to be too crude. In fact, already the simplest branching diffusions on bounded domains (3) suggest a more refined, though necessarily less general extension of Perron's theorem.

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

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