Some strong limit-2 and Dirichiet criteria for fourth order differential expressions

1990 ◽  
Vol 108 (2) ◽  
pp. 409-416 ◽  
Author(s):  
David Race
Keyword(s):  
Hilbert Space ◽  
Orthogonal Polynomials ◽  
Differential Expression ◽  
Fourth Order ◽  
Strong Limit ◽  
The Real ◽  
Image Position

AbstractConditions are given on the real coefficients p, q and r and the weight w, for the fourth order formally symmetric differential expressionto have the properties of being strong limit-2 and Dirichlet at ∞, when considered in the weighted Hilbert space, . These extend existing results due to both W. N. Everitt and V. Krishna Kumar and cover an expression which is important in the study of certain orthogonal polynomials.

On a fourth-order singular integral inequality

1978 ◽  
Vol 80 (3-4) ◽  
pp. 249-260 ◽  
Author(s):  
A. Russell
Keyword(s):  
Differential Expression ◽  
Fourth Order ◽  
Singular Integral ◽  
Integral Inequality ◽  
Second Order ◽  
New Order ◽  
Alternative Account ◽  
The Real ◽  
Image Position ◽  
Complete Solutions

SynopsisThe inequality considered in this paper iswhereNis the real-valued symmetric differential expression defined byGeneral properties of this inequality are considered which result in giving an alternative account of a previously considered inequalityto which (*) reduces in the casep=q= 0,r= 1.Inequality (*) is also an extension of the inequalityas given by Hardy and Littlewood in 1932. This last inequality has been extended by Everitt to second-order differential expressions and the methods in this paper extend it to fourth-order differential expressions. As with many studies of symmetric differential expressions the jump from the second-order to the fourth-order introduces difficulties beyond the extension of technicalities: problems of a new order appear for which complete solutions are not available.

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

Eigenvalues of Finite Band-Width Hilbert Space Operators and Their Application to Orthogonal Polynomials

1989 ◽  
Vol 41 (1) ◽  
pp. 106-122 ◽  
Author(s):  
Attila Máté ◽  
Paul Nevai
Keyword(s):  
Hilbert Space ◽  
Orthogonal Polynomials ◽  
Real Number ◽  
Main Diagonal ◽  
Band Width ◽  
Positive Part ◽  
Hilbert Space Operators ◽  
Image Position ◽  
Finite Band

The main result of this paper concerns the eigenvalues of an operator in the Hilbert space l2that is represented by a matrix having zeros everywhere except in a neighborhood of the main diagonal. Write (c)+ for the positive part of a real number c, i.e., put (c+ = cif c≧ 0 and (c)+=0 otherwise. Then this result can be formulated as follows. Theorem 1.1. Let k ≧ 1 be an integer, and consider the operator S on l2 such that

Dissipative Sturm-Liouville operators

1981 ◽  
Vol 88 (3-4) ◽  
pp. 329-343 ◽  
Author(s):  
Ian Knowles
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Minimal Operator ◽  
Sturm Liouville ◽  
Image Position ◽  
Complex Valued ◽  
Liouville Operators

SynopsisConsider the differential expressionwherepandw> 0 are real-valued andqis complex-valued onI. A number of criteria are established for certain extensions of the minimal operator generated by τ in the weighted Hilbert spaceto be maximal dissipative.

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.

Completeness properties in L2 of the eigenfunctions of two semi-linear differential operators

1980 ◽  
Vol 88 (3) ◽  
pp. 451-468 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Hilbert Space ◽  
Product Space ◽  
Orthonormal Basis ◽  
Differential Operators ◽  
Real Hilbert Space ◽  
The Real ◽  
Linear Differential Operators ◽  
Linear Problems ◽  
Linear Differential ◽  
Image Position

This paper concerns the boundary-value problemsin which λ is a real parameter, u is to be a real-valued function in C2[0, 1], and problem (I) is that with the minus sign. (The differential operators are called semi-linear because the non-linearity is only in undifferentiated terms.) If we linearize the equations (for ‘ small’ solutions u) by neglecting , there result the eigenvalues λ = n2π2 (with n = 1,2,…) and corresponding normalized eigenfunctionsand it is well known ((2), p. 186) that the sequence {en} is complete in that it is an orthonormal basis for the real Hilbert space L2(0, 1). We shall be concerned with possible extensions of this result to the non-linear problems (I) and (II), for which non-trivial solutions (λ, u) bifurcate from the trivial solution (λ, 0) at the points {n2π2,0) in the product space × L2(0, 1). (Here denotes the real line.)

On the Spectra of Unbounded Subnormal Operators

1986 ◽  
Vol 38 (5) ◽  
pp. 1135-1148 ◽  
Author(s):  
G. McDonald ◽  
C. Sundberg
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Real Line ◽  
Bounded Operator ◽  
Bounded Operators ◽  
The Real ◽  
Subnormal Operators ◽  
Hyponormal Operators ◽  
Image Position ◽  
Topological Boundary

Putnam showed in [5] that the spectrum of the real part of a bounded subnormal operator on a Hilbert space is precisely the projection of the spectrum of the operator onto the real line. (In fact he proved this more generally for bounded hyponormal operators.) We will show that this result can be extended to the class of unbounded subnormal operators with bounded real parts.Before proceeding we establish some notation. If T is a (not necessarily bounded) operator on a Hilbert space, then D(T) will denote its domain, and σ(T) its spectrum. For K a subspace of D(T), T|K will denote the restriction of T to K. Norms of bounded operators and elements in Hilbert spaces will be indicated by ‖ ‖. All Hilbert space inner products will be written 〈,〉. If W is a set in C, the closure of W will be written clos W, the topological boundary will be written bdy W, and the projection of W onto the real line will be written π(W),

On the location of eigenvalues of second order linear differential operators

1978 ◽  
Vol 80 (1-2) ◽  
pp. 15-22 ◽  
Author(s):  
Ian Knowles
Keyword(s):  
Boundary Condition ◽  
Hilbert Space ◽  
Differential Expression ◽  
Differential Operators ◽  
Homogeneous Boundary Condition ◽  
Upper Bounds ◽  
Order Linear ◽  
Linear Differential Operators ◽  
Linear Differential ◽  
Image Position

SynopsisThis paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expressionon [0,∞), together with a real homogeneous boundary condition at t = 0.

20.—Deficiency Indices of Polynomials in Symmetric Differential Expressions, II

1975 ◽  
Vol 73 ◽  
pp. 301-306 ◽  
Author(s):  
Anton Zettl
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Polynomial ◽  
Deficiency Indices ◽  
Linear Differential ◽  
Image Position ◽  
Necessary And Sufficient ◽  
The Relationship

SynopsisGiven a symmetric (formally self-adjoint) ordinary linear differential expression L which is regular on the interval [0, ∞) and has C∞ coefficients, we investigate the relationship between the deficiency indices of L and those of p(L), where p(x) is any real polynomial of degree k > 1. Previously we established the following inequalities: (a) For k even, say k = 2r, N+(p(L)), N−(p(L)) ≧ r[N+(L)+N−(L)] and (b) for k odd, say k = 2r+1where N+(M), N−(M) denote the deficiency indices of the symmetric expression M (or of the minimal operator associated with M in the Hilbert space L2(0, ∞)) corresponding to the upper and lower half-planes, respectively. Here we give a necessary and sufficient condition for equality to hold in the above inequalities.

XIII.—The Spectrum of a Fourth-Order Differential Operator

1970 ◽  
Vol 68 (3) ◽  
pp. 185-210
Author(s):  
Jyoti Chaudhuri ◽  
W. N. Everitt
Keyword(s):  
Differential Operator ◽  
Differential Expression ◽  
Fourth Order ◽  
Real Line ◽  
Alternative Methods ◽  
Order Differential Operator ◽  
The Real ◽  
Bounded Interval ◽  
Special Cases ◽  
Fourth Order Differential Operator

SynopsisThis paper considers the properties of the spectrum of a differential operator derived from differential expressions of the fourth order. With certain conditions on the coefficients of the differential expression the spectrum of the operator is discrete and an estimate is obtained of the number of eigenvalues lying in a given bounded interval of the real line. The results are compared with those obtained by alternative methods. Additional restrictions on the coefficients give special cases previously considered by other authors.

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