Discrete spectra criteria for singular differential operators with middle terms

1975 ◽  
Vol 77 (2) ◽  
pp. 337-347 ◽  
Author(s):  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Linear Operator ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Continuous Functions ◽  
Adjoint Extension ◽  
Image Position ◽  
Discrete Spectra ◽  
Bounded Below

Let l be the differential operator of order 2n defined bywhere the coefficients are real continuous functions and pn > 0. The formally self-adjoint operator l determines a minimal closed symmetric linear operator L0 in the Hilbert space L2 (0, ∞) with domain dense in L2 (0, ∞) ((4), § 17). The operator L0 has a self-adjoint extension L which is not unique, but all such L have the same continuous spectrum ((4), § 19·4). We are concerned here with conditions on the pi which will imply that the spectrum of such an L is bounded below and discrete.

The least limit point of the spectrum associated with sinǵular differential operators

1970 ◽  
Vol 67 (2) ◽  
pp. 277-281 ◽  
Author(s):  
M. S. P. Eastham
Keyword(s):  
Differential Operator ◽  
Linear Operator ◽  
Essential Spectrum ◽  
Limit Point ◽  
Compact Support ◽  
Complex Space ◽  
Differential Operators ◽  
Adjoint Extension ◽  
Image Position

Let τ be the formally self-adjoint differential operator denned bywhere the pr(x) are real-valued, , and p0(x) > 0. Then τ determines a real symmetric linear operator T0, given by T0f = τf, whose domain D(T0) consists of those functions f in the complex space L2(0, ∞) which have compact support and 2n continuous derivatives in (0, ∞) and vanish in some right neighbourhood of x = 0 ((7), p. 27–8). Since D(T0) is dense in L2(0, ∞), T0 has a self-adjoint extension T. We denote by μ the least limit point of the spectrum of T. The operator T may not be unique, but all such T have the same essential spectrum ((7), p. 28) and therefore μ does not depend on the choice of T.

Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)

1999 ◽  
Vol 129 (1) ◽  
pp. 165-179
Author(s):  
Yurii B. Orochko
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Differential Expression ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Image Position ◽  
Symmetric Differential Operators

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expressionacting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

Molchanov's Discrete Spectra Criterion for a Weighted Operator

1979 ◽  
Vol 22 (4) ◽  
pp. 425-431 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Hilbert Space ◽  
Continuous Functions ◽  
Second Order ◽  
Order Operator ◽  
Measurable Functions ◽  
Image Position ◽  
Discrete Spectra ◽  
Complex Valued ◽  
Finite Singularity

We consider the second-order operator1where the coefficients are real continuous functions on an interval ℐ with w and p positive. The operator is assumed singular at only one endpoint which we take to be either 0 (finite singularity) or ∞ (infinite singularity). Let be the Hilbert space of all complex-valued, measurable functions f satisfying

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.

The Two-Sided Factorization of Ordinary Differential Operators

1980 ◽  
Vol 32 (5) ◽  
pp. 1045-1057 ◽  
Author(s):  
Patrick J. Browne ◽  
Rodney Nillsen
Keyword(s):  
Differential Operator ◽  
Linear Function ◽  
Bilinear Form ◽  
Differential Operators ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
Ordinary Differential Operators ◽  
Image Position

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

scholarly journals Exhaustive operators and vector measures

1975 ◽  
Vol 19 (3) ◽  
pp. 291-300 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Vector Measures ◽  
Borel Functions ◽  
Image Position ◽  
Real Topological Vector Space ◽  
Sequence Of Functions

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 andthen Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representationwhere μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

scholarly journals On Dirichlet's boundary value problem for some formally hypoelliptic differntial operators

1988 ◽  
Vol 44 (3) ◽  
pp. 324-349 ◽  
Author(s):  
Niels Jacob
Keyword(s):  
Hilbert Space ◽  
Dirichlet Problem ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Differential Operators ◽  
Boundary Value ◽  
Divergence Form ◽  
Alternative Theorem ◽  
Sesquilinear Form ◽  
Homogeneous Dirichlet Problem

AbstractFor a class of formally hypoelliptic differential operators in divergence form we prove a generalized Gårding inequality. Using this inequality and further properties of the sesquilinear form generated by the differential operator a generalized homogeneous Dirichlet problem is treated in a suitable Hilbert space. In particular Fredholm's alternative theorem is proved to be valid.

Quasifree representations of Clifford algebras

1993 ◽  
Vol 113 (3) ◽  
pp. 487-497
Author(s):  
P. L. Robinson
Keyword(s):  
Hilbert Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Real Hilbert Space ◽  
Adjoint Operator ◽  
The State ◽  
Infinite Dimensional ◽  
Image Position

Let V be an infinite-dimensional real Hilbert space with associated C* Clifford algebra C[V]. To any state σ of the C* algebra C[V] there corresponds a skew-adjoint operator C of norm at most unity on V such thatwe refer to C as the covariance of the state σ.

scholarly journals Generalizations of the Aluthge transform of operators

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6465-6474 ◽  
Author(s):  
Khalid Shebrawi ◽  
Mojtaba Bakherad
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Convex Function ◽  
Polar Decomposition ◽  
Continuous Functions ◽  
Complex Hilbert Space ◽  
Numerical Radius ◽  
Aluthge Transform ◽  
Bounded Linear ◽  
Definition Of

Let A be an operator with the polar decomposition A = U|A|. The Aluthge transform of the operator A, denoted by ?, is defined as ? = |A|1/2U |A|1/2. In this paper, first we generalize the definition of Aluthge transformfor non-negative continuous functions f,g such that f(x)g(x) = x (x ? 0). Then, by using this definition, we get some numerical radius inequalities. Among other inequalities, it is shown that if A is bounded linear operator on a complex Hilbert space H, then h (w(A)) ? 1/4||h(g2 (|A|)) + h(f2(|A|)|| + 1/2h (w(? f,g)), where f,g are non-negative continuous functions such that f(x)g(x) = x (x ? 0), h is a non-negative and non-decreasing convex function on [0,?) and ? f,g = f (|A|)Ug(|A|).

A fixed point method to solve linear operator equations involving self-adjoint operators in Hilbert space

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3249-3251
Author(s):  
Mohammad Khan ◽  
Dinu Teodorescu
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Fixed Point Method ◽  
Operator Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Adjoint Operators ◽  
Invertible Matrices ◽  
Linear Operator Equations

In this paper we provide existence and uniqueness results for linear operator equations of the form (I+Am) x = f , where A is a self-adjoint operator in Hilbert space. Some applications to the study of invertible matrices are also presented.

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