Spectral Operators Of Scalar Type Research Articles

The non-equivalence of pseudo-Hermiticity and presence of antilinear symmetry

The non-equivalence of the presence of antilinear symmetry and pseudo-Hermiticity is shown for bounded operators. Two appropriate examples are operators with non-empty residual spectrum. The class of operators for which the equivalence holds is extended to the spectral operators of scalar type. The importance of J-self-adjointness is stressed and new proofs using this property are provided.

A criterion for Hill operators to be spectral operators of scalar type

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schro dinger operator) H = −d 2 /dx 2 + V to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. In the course of our analysis, we also establish a functional model for periodic Schrodinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion. The problem of deciding which Hill operators are spectral operators of scalar type appears to have been open for about 40 years.

When is a non-self-adjoint Hill operator a spectral operator of scalar type?

We derive necessary and sufficient conditions for a one-dimensional periodic Schrödinger (i.e., Hill) operator H = − d 2 / d x 2 + V in L 2 ( R ) to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. To cite this article: F. Gesztesy, V. Tkachenko, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

A note on the spectral operators of scalar type and semigroups of bounded linearoperators

It is shown that, for the spectral operators of scalar type, the well‐known characterizations of the generation of C0‐ and analytic semigroups of bounded linear operators can be reformulated exclusively in terms of the spectrum of such operators, the conditions on the resolvent of the generator being automatically met and the corresponding semigroup being that of the exponentials of the operator.

On an abstract evolution equation with a spectral operator of scalar type

It is shown that the weak solutions of the evolution equation y′(t) = Ay(t), t ∈ [0, T) (0 < T ≤ ∞), where A is a spectral operator of scalar type in a complex Banach space X, defined by Ball (1977), are given by the formula y(t) = e tAf, t ∈ [0, T), with the exponentials understood in the sense of the operational calculus for such operators and the set of the initial values, f′s, being ∩ 0≤t<TD(e tA), that is, the largest possible such a set in X.

Periodic groups of operators in Banach spaces

Spectral operators of scalar type in the sense of Dunford often occur in connection with unconditionally convergent series expansions, whereas conditionally convergent expansions under similar conditions may be described with the help of operators having a more general type of spectral decomposition. We show that under certain conditions even in the latter case we can restrict our considerations to a dense linear submanifold of the original Banach space with a stronger topology, where the convergence of the expansion under study will be unconditional. Though our conditions could be formulated in terms of a single operator, it seems to be more natural to state them in terms of (the generator of) a periodic group of operators.

A note on U-scalar operators

U-scalar operators are discussed in this paper. We give the necessary and sufficient conditions for an operator with purely discrete spectrum to be u-scalar. And generally, we prove that a u-scalar operator is Hermitian in the sense of some topology. A new example of u-scalar operator is given. This operator is not a spectral operator of scalar type.

Regularized scalar operators

We introduce a C-regularized scalar operator. These have the properties of spectral operators of scalar type, except that the spectral measure is bounded and countably additive only after applying the regularizing operator C. We discuss the relationship between regularized scalar operators, regularized functional calculi, and generating a polynomially bounded regularized group.

The Semi-simplicity Manifold on Arbitrary Banach Spaces

For an arbitrary linear (possibly unbounded) operator A on a Banach space, with real spectrum, we construct a maximal continuously embedded Banach subspace on which this operator has a Cℓ(R) functional calculus. We call this subspace, Z, the semi-simplicity manifold for A. When the original Banach space does not contain a copy of c0, the restriction of A to Z is a spectral operator of scalar type. We construct a functional calculus, f ↦ f(A |Z), from C(R) into the space of closed, densely defined operators on Z; when X does not contain a copy of c0, this map is defined for arbitrary Borel measurable f. We also construct continuously embedded Banach subspaces on which the Fourier transform and the Hilbert transform are spectral operators of scalar type.

Lie and Jordan Ideals in B(c 0 ) and B(l ρ )

It is shown that ideals with respect to the canonical Lie (commutator) product in these algebras are exactly the linear manifolds that contain the images of their elements under the action of inner automorphisms induced by invertible spectral operators of scalar type. Jordan ideals in these algebras are identical with two-sided associative ideals and are also applied to a characterization of Lie ideals. It is well known that any associative algebra A becomes a Lie algebra and a Jordan algebra under the products defined by [F, G] = FG GF and F o G = '(FG + GF) for F, G E A, respectively. (The ground field is C.) A linear manifold L in A is called a Lie (Jordan) ideal if the corresponding (Lie or Jordan) product belongs to L for every F E A and G E L. It is then a two-sided ideal with respect to the corresponding product. Similarly, by an associative ideal we shall understand a two-sided ideal under the associative multiplication. Remarkable characterizations of the (not necessarily closed) Lie and Jordan ideals in the case when H is a complex infinite-dimensional separable Hilbert space and A is B(H), the (associative) algebra of all bounded linear operators in H, were described by Fong, Miers, and Sourour [3; Theorems 1 and 3] and obtained partly by Topping (see [3, 11]). Fong and Murphy [4] showed that [3, Theorem 1] is valid in the nonseparable case, too. The purpose of this paper is to prove that [3, Theorems 1 and 3] can be extended in a suitable form for the case when the underlying complex Banach space is either co or Ip (1 < p < xo) . The proofs are based on some fundamental results of Pelczynski [9] on complemented subspaces and isomorphisms in the Banach spaces above. Owing to them, we can make use of several ideas, applied earlier in the case of a separable Hilbert space, which we shall not reproduce here in detail. For a reference on spectral operators see, e.g., the book of Dunford and Schwartz [1]. For a bounded linear operator in a Banach space X, ker and im will denote its kernel and range subspace, respectively. An idempotent will be called any operator P E B(X) satisfying p2 = P. The notations (X E X D ... )x and Received by the editors June 17, 1991. 1991 Mathematics Subject Classification. Primary 47D30, 47B40, 17B60.

Lie and Jordan ideals in 𝐵(𝑐₀) and 𝐵(𝑙ᵨ)

It is shown that ideals with respect to the canonical Lie (commutator) product in these algebras are exactly the linear manifolds that contain the images of their elements under the action of inner automorphisms induced by invertible spectral operators of scalar type. Jordan ideals in these algebras are identical with two-sided associative ideals and are also applied to a characterization of Lie ideals.

Fourier transforms and the dilatation group

The Fourier transform g(k) of a square integrable function f(x), vanishing for x&amp;lt;0, is analytic in the upper half plane, so that, replacing k by k exp ζ, k≥0, 0&amp;lt;Im ζ&amp;lt;π, it can be associated with an operator K(ζ) in ℋ+=L2((0,∞),dx). The operator K(ζ) can be expressed in terms of the generator D of the dilatation group on ℋ+ and it can be shown that it is analytic in the strip 0&amp;lt;Im ζ&amp;lt;π with strong limits as Im ζ↓0 and ↑π. The Laplace transform (ζ=iπ/2) is an analytic vector for D. It is also found that D is not a spectral operator of scalar type on Lp((0,∞),dx), 1≤p&amp;lt;∞, p≠2. Applying the results obtained here to the time-evolution operator for a one-dimensional Sommerfeld model for the interaction between an electron and a metal, it is found that this operator has a complex-dilated analytic extension.

Cauchy problems involving non-selfadjoint operators

We consider the differential equation where for each is f(t) an element of a Hilbert space H and H: dom H → H is a closed, densely defined linear operator with real spectrum. We present a class, the U-scalar operators, which possess a representation of the form on a dense set. Here E(S), s∊R is a family of possibly unbounded commuting projections. U-scalar operators are a generalization of spectral operators of scalar type. If H is U-scalar we show that defines a one parameter group of denselydefined linear operators and that, for a dense set of initial conditions f0, a strong solution of (1) is given by f(t)=B(t)f0. We give necessary and sufficient conditions for an operator to be U-scalar. We prove that U-scalar operators do not possess residual spectrum

A note on the spectral theorem

We give necessary and sufficient conditions to prove a spectral theorem and a functional calculus for certain nonselfadjoint operators, H. Our method is non-perturbative: the conditions are given in terms of the resolvent (z-H)−1. We give an example of an operator satisfying these conditions. This operator is not a spectral operator of scalar type. Its spectral projections are unbounded operators defined on a common dense domainD.

Spectral Operators of Scalar Type in Grothendieck Spaces with the Dunford-Pettis Property

Spectral Operators of Scalar Type in Grothendieck Spaces with the Dunford-Pettis Property