scholarly journals On a Class of Non-Self-Adjoint Differential Operators

1960 ◽  
Vol 12 ◽  
pp. 641-659 ◽  
Author(s):  
R. R. D. Kemp
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Spectral Analysis ◽  
Differential Operator ◽  
Spectral Measure ◽  
Differential Operators ◽  
Expansion Theorem ◽  
Completely Continuous ◽  
Normal Operators ◽  
Uniformly Bounded

The problem of spectral analysis of non-self-adjoint (and non-normal) operators has received considerable attention recently. Livšic (5), and more recently Brodskii and Livšic (1) have considered operators on Hilbert space with completely continuous imaginary parts. Dunford (3) has generalized the notion of spectral measure and defined a class of spectral operators on Hilbert and Banach space. Schwartz (8) and Rota (7) have investigated conditions under which a differential operator will be spectral. The work of Naimark (6) and the author (4) on non-self-adjoint differential operators leads to an expansion theorem which implicitly defines a type of spectral measure. However the projections involved in this will not in general be bounded, much less uniformly bounded.

scholarly journals TENSOR EXTENSION PROPERTIES OF C(K)-REPRESENTATIONS AND APPLICATIONS TO UNCONDITIONALITY

2010 ◽  
Vol 88 (2) ◽  
pp. 205-230 ◽  
Author(s):  
CHRISTOPH KRIEGLER ◽  
CHRISTIAN LE MERDY
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unconditional Basis ◽  
Compact Set ◽  
Bounded Operators ◽  
Space Setting ◽  
Uniformly Bounded

AbstractLet K be any compact set. The C*-algebra C(K) is nuclear and any bounded homomorphism from C(K) into B(H), the algebra of all bounded operators on some Hilbert space H, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept ofR-boundedness. Then we apply these results to operators with a uniformly bounded H∞-calculus, as well as to unconditionality on Lp. We show that any unconditional basis on Lp ‘is’ an unconditional basis on L2 after an appropriate change of density.

scholarly journals Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

Extension of polynomials defined on subspaces

2010 ◽  
Vol 148 (3) ◽  
pp. 505-518 ◽  
Author(s):  
MAITE FERNÁNDEZ-UNZUETA ◽  
ÁNGELES PRIETO
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Homogeneous Polynomial ◽  
Analogous Result ◽  
Holomorphic Functions ◽  
Bounded Norm ◽  
Uniformly Bounded

AbstractLet k ∈ ℕ and let E be a Banach space such that every k-homogeneous polynomial defined on a subspace of E has an extension to E. We prove that every norm one k-homogeneous polynomial, defined on a subspace, has an extension with a uniformly bounded norm. The analogous result for holomorphic functions of bounded type is obtained. We also prove that given an arbitrary subspace F ⊂ E, there exists a continuous morphism φk, F: (kF) → (kE) satisfying φk, F(P)|F = P, if and only E is isomorphic to a Hilbert space.

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 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.

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 On commutative V*-algebras

1970 ◽  
Vol 17 (2) ◽  
pp. 173-180 ◽  
Author(s):  
P. G. Spain
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Boolean Algebra ◽  
Unitary Groups ◽  
Easy Proof ◽  
Complete Extension ◽  
Normal Operators ◽  
Similar Extension ◽  
Stone’S Theorem

We shall use results of Palmer (10, 11) and of Edwards and Ionescu Tulcea (6) to show that a commutative V*-algebra (with identity) of operators on a weakly complete Banach space is isomorphic to such an algebra on a Hilbert space, the isomorphism extending to the weak closures of the algebras. This result leads to an extension of Stone's theorem on unitary groups (a similar extension is proved by different methods in (2, p. 350) and of Nagy's theorems on semigroups of normal operators. The same technique yields an easy proof of Dunford's theorem on the existence of a σ-complete extension of a bounded Boolean algebra of projections on a weakly complete Banach space. We are indebted to H. R. Dowson for suggesting this topic and for help and guidance in pursuing it.

On operators of class (N, k)

1970 ◽  
Vol 68 (1) ◽  
pp. 141-142 ◽  
Author(s):  
P. B. Ramanujan
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Transformation ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Normal Operators ◽  
Bounded Linear Transformation

Istrăţescu (2) has introduced a class of operators on a Banach space called operators of class (N, k). An operator T (a bounded linear transformation) on a Banach space X is said to be an operator of class (N, k), k = 2, 3,…, if ‖Tx‖k ≤ ‖Tkx‖ for all x ∈ X such that ‖x‖ = 1. If k = 2, such an operator is called an operator of class (N)(3). If X is a Hilbert space, then the class of operators of class (N) on X is an extension of the class of hyponormal operators on X (3). The object of this note is to generalize to operators of class (N, k) on a Banach space X, some results which are known to be true for normal operators on a Hilbert space, particularly with regard to their ascent and descent.

scholarly journals Onnth-order differential operators with Bohr-Neugebauer type property

1987 ◽  
Vol 10 (1) ◽  
pp. 51-55
Author(s):  
Aribindi Satyanarayan Rao
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Operator ◽  
Differential Operators ◽  
Bounded Solution ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Bounded Linear ◽  
Periodic Continuous Function ◽  
Type Property

SupposeBis a bounded linear operator in a Banach space. If the differential operatordndtn−Bhas a Bohr-Neugebauer type property for Bochner almost periodic functions, then, for any Stepanov almost periodic continuous functiong(t)and any Stepanov-bounded solution of the differential equationdndtnu(t)−Bu(t)=g(t),u(n−1),…,u′,uare all almost periodic.

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