On some Operator Functional Equations in Locally Convex Algebras

This paper aims to prove the existence of the solutions of some operator functional equations of sinus type in universally bounded operator algebra, where the operator is defined on a locally convex space. Some results based on the work of B. Sz.-Nagy following the representation model of operator groups are established.

A completely new area where applied research could be the solution to various unresolved issues from the engineering sciences, starting with the equations of motion of atomic particles up to complex signals theories is represented by operator functional equations within a larger framework: locally convex spaces.
This research is connected with the study of functional equations of sine and cosine in different algebraic-topological structures, as well as in the locally convex algebra of universally bounded operators defined on a locally convex space X, in connection with the theory of semigroups (groups) of operators. In this respect, there could be mentioned that a first step in this direction was made by S. Kurepa [14] obtaining a generalization of equations that characterize the trigonometric and hyperbolic functions by functional equations of matrices. World first works in operatorial trigonometric functional equations belong to S. Kurepa [14] and H.O. Fattorini [8].
In 1958, V. Ptak in [20] introduces the B− completeness of a locally convex space that allows an extension of the fundamental principles of functional analysis to such spaces. For such an extension, a result of D. Van Dulst [5] regarding the heredity property of a barreled space is needed.
Based on some known results obtained by Allan [1], F. G. Bonales and R. V. Mendoza [2], develop a version of spectral theory for bounded linear operators on barreled spaces.
All locally convex spaces will be assumed Hausdorff and over the complex field C. A calibration for the locally convex space X is a family P of semi-norms generating its topology, in the sense that the topology of X is the coarsest with respect to which all the semi-norms in P are continuous. A calibration P is characterized by the property that the sets E(p,ϵ)={xϵX: p(x)<ϵ}, ϵ>0, pϵP constitute a neighborhood sub-base at 0. (see [6, 7, 11]) We denote by (X,P) a locally convex Hausdorff space with a calibration P. A locally convex algebra is algebra with a locally convex topology in which the multiplication is separately continuous. Such an algebra is locally m-convex (l.m.c.) if it has a neighborhood base U at 0 such that U ∈ U is convex, balanced (λU ⊆ U for |λ| < 1) and satisfies the semigroup property U 2 ⊆ U.
Any algebra with identity will be called unital. A unital l.m.c. algebra A is characterized by the existence of a calibration P such that each p∈P is submultiplicative (p(xy)≤p(x)p(y)) and satisfies p(e)=1, where e is the unit element. A unital lmc algebra A is characterized by the existence of a calibration P such that each pϵ P is sub-multiplicative (p(xy)≤ p(x)p(y)) and satisfies p(1)=1. We recall that the dual X * of a locally convex space X is endowed with the topology of uniform convergence on finite subsets of X, denoted by σ(X * , X). We also mention that U 0 denotes the polar of U ( [11, 12 and 13]). According to V. Ptak [20], a locally convex space X is B-complete if a linear sub-space Y⊂ X * is σ(X * , X) closed whenever Y∩U 0 is closed for each 0-neighborhood U⊂ X. Let A be a complex algebra of functions, like in [18]. By a locally multiplicative convex algebra (l.m.c.) we understand a topologically algebra whose topology is given by the family of sub-multiplicative semi-norms {p α }, i.e. p α (xy)≤ p α (x) p α (y), for any x, yϵA and αϵI. An algebra A is called proper if only the zero element annihilates the whole algebra A, i.e. if xA=Ax={0} then x=0. It is obvious that any semi-simple algebra is proper. Let A be a semi-simple commutative algebra. Then from [10] a multiplier is a mapping T: A → A which verifies Tx· y=x· Ty, for all x,yϵA.
We denote the set of all multipliers on A by M(A). We observe that the identity x · Ty = T(xy) = Tx · y holds for some x,yϵA and thus the range TA and the kernel ker T of T are both bilateral ideals of A (see [15,17]).
By G. R. Allan [1], an element x ∈ X, where X is a locally convex algebra, is said to be bounded in X if there exists α∈C such that the set {(αx) n } n≥1 is bounded in X. The set of all bounded elements in X will be denoted by X 0 .
We call resolvent set in the Waelbroeck sense of an element x from a locally convex unital algebra (X,P) the set of all elements λ 0 ∈ C ∞ (C∪{∞}) for which there exists V∈V λ0 such that the following conditions hold: (a) the element λe−x is invertible in X, for any λ∈V\{∞}; The resolvent set will be denoted by ρ(x). We also denote by

Introductory notions
Functional equations, both classic and operator ones were born from mathematical physics or from functional characterization of elementary functions. D'Alembert in 1769 obtains cosine equation with 3 unknown functions in 1804, starting with composition of forces, Poisson reaches to the cosine functional equation (2.2) In 1909 D.R. Carmichael determines the analytic solutions of sine functional equation In the following we will present sine and cosine functional equations in a different algebraic-topological structure, namely in an l.m.c. algebra. First papers on the subject of trigonometric operator functional equations worldwide belong to S. Kurepa [14] and H.O. Fattorini [8].
It is known that if X is an arbitrary locally convex space, then the function f:R->X is measurable if there are a number of simple functions that converges a.e. on R to f (t). The function f is strongly continuous if the condition t->t 0 implies f (t)->f (t 0 ) in the topology given by the norm of X. The function g: R->B(X) is called operator function. It is continuous: if t -> t 0 implies g(t) ->g(t 0 ) in the topology given by the norm of B(X) and strongly continuous if for any xϵX, the function f (t)=g(t)x is continuous.
Let If X is a complex Banach space and f is an operatorial strongly measurable function which satisfies f (0) =I (I being the identity operator), then f is called cosine operatorial function. With

From 1), 2) and hypothesis (2.4) it results 3). By replacing f(s·t -1 )=2f(s)f(t)-f(s·t) in 3), it results 4). To prove B f is commutative one can inter-change s with t and we get f(t·s)+f(t·s -1 )=2f(t)f(s), for all t,sϵG. From this and from (2.1), based on conditions (2.4) it follows f(s)f(t)=f(t)f(s) for every s,tϵG, equality which shows that sub-algebra B f is commutative. By taking t=e in (2.1) we get f(s)=f(s)f(e).
Because the sub-algebra is commutative it will result f(e)=I f is the unit element from .
In the following we will prove that if h: (G, ·)→B is a homeo-morphism of the multiplicative group (G, ·) on a multiplicative sub-group of B:

h(s·t)=h(s)h(t) with the condition h(s·t) =h(t·s) for every s,tϵG then the function f: (G, ·)→B defined by f(s)=[h(s)+h(s -1 )]/2
(2.5) is a solution of (2.1). So, the following two representation theorems of the cosine functional equation solutions hold.

Theorem 2.2 If h: (G, ·) → B is a homomorphism of the group (G, ·) on a multiplicative sub-group of X and if h(s · t) = h(t · s)
, for all s, t ∈ G then the function defined by (2.5) is a solution of (2.1).

Proof:
We have to show that the function f given by (2.5) with the help of homo-morphism h, verifies equation (2.1). This can be deduced from the sequence: 2f It remains to prove h is a homo-morphism of G. To do that, from definition of h (2.7) we have: