On Necessary Conditions for Scalars

In this paper, we give a characterization of scalar operators. In particular we show that a densely defined closed linear operator H acting on a reflexive Banach space X is scalar if it is of (0,1) type R and � f(H) �≤� f � ∞ for f in the algebra of smooth functions U.


Introduction
Suppose H is a closed densely defined operator on a Banach space X, whose spectrum is contained in R and there exist a C > 0 such that for all z ∈ iR and some (α, α + 1) ≥ 0 then H is of (α, α + 1) type R [1].
A special case is a Hermitian operator on a Hilbert space.Scalar operators with real spectrum is called a pseudo-hermitian operator.In Hilbert space, abounded linear operator S is a pseudo-hermitian if and only if the group e itS ≤ M < ∞ for all t ∈ R [8].If X is a reflexive Banach space then an operator T ∈ B(X) is scalar spectral if it admits an integral representation with respect to countably additive projection valued measure or equivalently if it admits a C(σ(T )) functional calculus [6].In particular, if T acts on a Hilbert space H, then T admits C(R) functional calculus if it is Hermitian.Generally, an operator acting on a reflexive Banach space is scalar if and only if it has a C o (R) functional calculus [5].According to [10], If T is an operator with σ(T ) ⊂ R and acting on a reflexive Banach space X, then T is scalar if and only if iH generates a uniformly bounded strongly continuous group.In [7], a functional calculus is given for a closed densely defined linear operators on a Banach space with σ(H) ⊆ R satisfying the resolvent estimate and for functions from weighted sobolev spaces.Here the calculus used is based on almost analytic extension to C of infinitely differentiable functions defined on R and the Helffer-Sjostrand formula [9].Such calculus defines an algebra homomorphism.We now consider an intermediate topology is the space of smooth functions of compact support.For detailed information see [2].For any f ∈ U the norm is defined as; where for all x, β ∈ R and c r > 0.
It is shown in [2] that U is an algebra under pointwise multiplication.The definition of f (H) for f ∈ U originates from the version of Helffer and Sjöstrand [9] integral formula.Using this and the abstract result from [3], we show that a densely defined closed linear operator H acting on a reflexive Banach space X is scalar if it is of (0, 1) type R and f (H) ≤ f ∞

The U functional Calculus
The materials in this section has been taken from [3] and [4].
For any f ∈ U and n ≥ 0 an almost analytic extension of f to C is defined; where and it is proved in [3], that for n > α ≥ 0 and doesn't depend on τ ; • the mapping extends to a bounded algebra homomorphism; • if f ∈ U and f = 0 on a neighbourhood of σ(H) then f (H) = 0; For an operator H of (α, α + 1)-type R, we associate each element f ∈ U with an operator f (H) ∈ B(X) given by (6) In order to state our results, we need the following theorems and corollaries; Theorem 2.1 Let H be a bounded operator with σ(H) ⊆ R, and In particular H is a pseudo hermitian operator, and so it is a scalar operator.

Theorem 2.3 H is a generator of a C o -contraction semi-group if and only if H is closed, densely defined and for each
one parameter group T .
Theorem 2.9 A densely defined linear operator H acting on a reflexive Banach space X, is scalar if it is of (0, 1)-type R and f (H) ≤ f ∞ for each f ∈ U Proof.Let H be an operator acting on Hilbert space H and σ(H) ⊆ R, then H is A Pseudo Hermitian Operator.By theorem (2.5) it is of (0,1)-type R and by theorem (2.6) it is a scalar operator.Also iH generates a one parameter group by theorem 2.8, hence H admits a functional calculus given by (6).Since ( 6) is continuous by ( 1) and ( 5), the resolvent set is bounded.From theorem (2.7) we see that (6) converges absolutely.Since H is Hermitian, it follows by Riesz representation theorem that for f ∈ U there exist a complex Borel measure μ on σ(H) such that f (H) = σ(H) f (z)μdz and this completes the proof.