scholarly journals Spectrums of Solvable Pantograph Differential-Operators for First Order

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Z. I. Ismailov ◽  
P. Ipek
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Operator Theory ◽  
Differential Operators ◽  
Finite Interval ◽  
Exact Formula ◽  
Operator Expression ◽  
Minimal Operator ◽  
First Order ◽  
Delay Differential

By using the methods of operator theory, all solvable extensions of minimal operator generated by first order pantograph-type delay differential-operator expression in the Hilbert space of vector-functions on finite interval have been considered. As a result, the exact formula for the spectrums of these extensions is presented. Applications of obtained results to the concrete models are illustrated.

Reducibility of differential operators some: examples

1979 ◽  
Vol 86 (1) ◽  
pp. 29-33 ◽  
Author(s):  
B. Fishel ◽  
N. Denkel
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Volterra Operator ◽  
Second Order ◽  
Order Differential Operator ◽  
Order Operator ◽  
Minimal Operator ◽  
Deficiency Indices ◽  
First Order ◽  
Symmetric Operators

A symmetric operator on a Hilbert space, with deficiency indices (m; m) has self-adjoint extensions. These are ‘highly reducible’. The original operator may be irreducible, (see example (i), below). Can the mechanism whereby reducibility is achieved be understood? The concrete examples most readily studied are those associated with differential operators. It is easy to obtain operators, associated with a formal linear differential operator, having deficiency indices (m; m). What of reducibility? Nothing seems to be known. In the case of the first-order operator we were able, using the Volterra operator, to establish irreducibility of the associated minimal operator. To investigate symmetric operators associated with a second-order differential operator, different methods had to be developed. They apply also to the first-order operator, and we employ them to demonstrate the irreducibility of the associated minimal operator. In the second-order case the minimal operator proves reducible, and we also exhibit examples of reducibility of associated symmetric operators. It would clearly be of interest to elucidate the influence of the boundary conditions on reducibility.

First Order Operators on Manifolds With a Group Action

1996 ◽  
Vol 48 (4) ◽  
pp. 758-776 ◽  
Author(s):  
H. D. Fegan ◽  
B. Steer
Keyword(s):  
Differential Operator ◽  
Group Action ◽  
Lie Group ◽  
Differential Operators ◽  
Order Differential Operator ◽  
Compact Lie Group ◽  
First Order ◽  
Rank 2 ◽  
Homogeneous Operators

AbstractWe investigate questions of spectral symmetry for certain first order differential operators acting on sections of bundles over manifolds which have a group action. We show that if the manifold is in fact a group we have simple spectral symmetry for all homogeneous operators. Furthermore if the manifold is not necessarily a group but has a compact Lie group of rank 2 or greater acting on it by isometries with discrete isotropy groups, and let D be a split invariant elliptic first order differential operator, then D has equivariant spectral symmetry.

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.

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.

scholarly journals A HOLOMORPHIC REPRESENTATION OF THE JACOBI ALGEBRA

2006 ◽  
Vol 18 (02) ◽  
pp. 163-199 ◽  
Author(s):  
STEFAN BERCEANU
Keyword(s):  
Hilbert Space ◽  
Differential Operators ◽  
Holomorphic Functions ◽  
First Order ◽  
Polynomial Coefficients ◽  
Holomorphic Representation

A representation of the Jacobi algebra 𝔥1 ⋊ 𝔰𝔲(1, 1) by first-order differential operators with polynomial coefficients on the manifold [Formula: see text] is presented. The Hilbert space of holomorphic functions on which the holomorphic first-order differential operators with polynomials coefficients act is constructed.

scholarly journals Ger-type and Hyers-Ulam stabilities for the first-order linear differential operators of entire functions

2004 ◽  
Vol 2004 (22) ◽  
pp. 1151-1158 ◽  
Author(s):  
Takeshi Miura ◽  
Go Hirasawa ◽  
Sin-Ei Takahasi
Keyword(s):  
Entire Function ◽  
Differential Operator ◽  
Stability Problem ◽  
Differential Operators ◽  
Entire Functions ◽  
Ulam Stability ◽  
Nonzero Constant ◽  
First Order ◽  
Linear Differential Operators ◽  
Linear Differential

Lethbe an entire function andTha differential operator defined byThf=f′+hf. We show thatThhas the Hyers-Ulam stability if and only ifhis a nonzero constant. We also consider Ger-type stability problem for|1−f′/hf|≤ϵ.

scholarly journals Involutive Property of Resolutions of Differential Operators

1966 ◽  
Vol 27 (2) ◽  
pp. 419-427
Author(s):  
Masatake Kuranishi
Keyword(s):  
Vector Bundle ◽  
Differential Operator ◽  
Vector Space ◽  
Cross Sections ◽  
Vector Bundles ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
First Order ◽  
Linear Differential

Let E and E′ be C∞ vector bundles over a C∞ manifold M. Denote by Γ(E) (resp. by Γ(E′) the vector space of C∞ cross-sections of E (resp. of E′) over M. Take a linear differential operator of the first order D: Γ(E) → Γ(E′) induced by a vector bundle mapping σ(D): jl(E) ′ E′, where Jk(E) denotes the vector bundle of k-jets of cross-sections of E.

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.

scholarly journals Singularities of the eta function of first order differential operators

2012 ◽  
Vol 20 (2) ◽  
pp. 59-70
Author(s):  
Paul Loya ◽  
Sergiu Moroianu
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Closed Manifold ◽  
Order Differential Operator ◽  
First Order ◽  
Positive Integers

Abstract We report on a particular case of the paper [7], joint with Raphaël Ponge, showing that generically, the eta function of a first-order differential operator over a closed manifold of dimension n has first-order poles at all positive integers of the form n - 1; n - 3; n - 5;. . . .

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