On strang’s fifth bound for eigenvalues of the jordan product of two self-adjoint operators on a finite dimensional unitary space

1989 ◽  
Vol 15 (1-2) ◽  
pp. 53-65
Author(s):  
A. Grubb ◽  
C. S. Sharma
Keyword(s):  
Unitary Space ◽  
Jordan Product ◽  
Finite Dimensional ◽  
Adjoint Operators

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.

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.

Self-adjoint operators on inner product spaces over fields of power series

2008 ◽  
Vol 58 (4) ◽  
Author(s):  
Hans Keller ◽  
Ochsenius Herminia
Keyword(s):  
Power Series ◽  
Adjoint Operator ◽  
Inner Product ◽  
Closed Field ◽  
Product Spaces ◽  
Binary Forms ◽  
Inner Product Spaces ◽  
Finite Dimensional ◽  
Orthogonal Decompositions ◽  
Adjoint Operators

AbstractTheorems on orthogonal decompositions are a cornerstone in the classical theory of real (or complex) matrices and operators on ℝn. In the paper we consider finite dimensional inner product spaces (E, ϕ) over a field K = F((χ 1, ..., x m)) of generalized power series in m variables and with coefficients in a real closed field F. It turns out that for most of these spaces (E, ϕ) every self-adjoint operator gives rise to an orthogonal decomposition of E into invariant subspaces, but there are some salient exceptions. Our main theorem states that every self-adjoint operator T: (E, ϕ) → (E, ϕ) is decomposable except when dim E is a power of 2 with exponent at most m, and ϕ is a tensor product of pairwise inequivalent binary forms. In the exceptional cases we provide an explicit description of indecomposable operators.

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 Spectral Enclosures and Complex Resonances for General Self-Adjoint Operators

1998 ◽  
Vol 1 ◽  
pp. 42-74 ◽  
Author(s):  
E.B. Davies
Keyword(s):  
Spectral Properties ◽  
Linear Subspace ◽  
Adjoint Operator ◽  
Dimensional Linear Subspace ◽  
Finite Dimensional ◽  
Adjoint Operators ◽  
Computation Of Eigenvalues

AbstractThis paper considers a number of related problems concerning the computation of eigenvalues and complex resonances of a general self-adjoint operator H. The feature which ties the different sections together is that one restricts oneself to spectral properties of H which can be proved by using only vectors from a pre-assigned (possibly finite-dimensional) linear subspace L.

Finite-dimensional perturbations of self-adjoint operators

1999 ◽  
Vol 34 (2) ◽  
pp. 127-164 ◽  
Author(s):  
Jonathan Arazy ◽  
Leonid Zelenko
Keyword(s):  
Finite Dimensional ◽  
Adjoint Operators

QUADRATIC EQUATIONS IN HILBERTIAN OPERATORS AND APPLICATIONS

2009 ◽  
Vol 20 (11) ◽  
pp. 1431-1454
Author(s):  
VICTOR J. MIZEL ◽  
M. M. RAO
Keyword(s):  
Random Fields ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Infinite Dimensions ◽  
Bounded Linear ◽  
Quadratic Equations ◽  
Weak Ordering ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Adjoint Operators

In this paper bounded linear operators in Hilbert space satisfying general quadratic equations are characterized. Necessary and sufficient conditions for sets of operators satisfying two such equations to compare relative to a weak ordering are presented. In addition, averaging operators in finite dimensional spaces are determined, and in this case it is shown that they are unitary models for all projections. It is pointed out, by an example, that the latter result does not hold in infinite dimensions. A key application to certain second order random fields of Karhunen type is given. The main purpose is to present the structure of bounded non-self adjoint operators solving quadratic equations, and indicate their use.

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