On a calculus of operators in reflexive vector spaces

Abstract

Hubert space.*The problem of characterizing in simple fashion operators which may be resolved is still open.It may be of interest to note that preliminary attempts in that direction incline one to an optimistic outlook. II. The algebra of projections1.The space.We operate in a normed linear vector space 58 whose elements will be designated by/, g, h, .Addition of elements and multiplication by complex numbers (denoted by p, a-, t, ) is permitted subject to the customary restrictions.The norm of/, ||/||, is a real-valued function which satisfies the conditions ||/||^0, ||/||=0 implies /=0, ||p/|| = |p|||/||, and ||/+g|| 11/11 +||g||-If we write distance (/, g) = \\f-g\\, the norm metrizes 58.If the sequence {/"} converges to / in this metric, we often write /"-/ for ||/-/||-*0.The space 58 is assumed to be complete in this metricf An operation O is a function whose domain is the space 58 and whose range is a subset of a space of the same type 58i.O is said to be distributive if 0(pf+ag)=pOf+aOg.If O is distributive and continuous at every point of 58, then O is said to be linear.It is known that if O is linear (and only then if O is distributive), there exists a constant A^O independent of / e 58, such that \\Of\\ K\\f\\.The least such constant K is called the bound of 0. The operation O is said to be closed if /"-/, Qfn-*g implies Of=g.A closed distributive operation is linear.An operation whose range is contained in the initial space 58 is called an operator or transformation.Operators are denoted by A, B, P, Q, .Two special operators and their defining equations are 0,1, with 0/=0, 7/=/.The bounds of A, P, are written | A |, | P |, .An operation whose range is a set of complex numbers is called a functional.Functionals will be denoted by F, G, .The totality of linear functional defined on 58 is a linear set.If we write |P|| for the bound of F, the set of functionals is a complete space (58) of the same type as 58.The space ( 58) is said to be the space adjoint to 58.The space ((58)) may now be introduced in evident fashion.If A is a linear operator, and F is a linear functional, both defined on 58, then F (A) is a linear functional.Indeed, F(A[f+g])=F(Af+Ag)=F(Af) +F(Ag).Also * For recent developments in the theory of the operational calculus in Hilbert space cf.