The essential state diagram of a linear operator
R. W. Cross · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1990
Given an operator T: D(T) C X --+ I/ where X and Y are normed spaces, we cal| T on tr,+ -operator [C2] if there exists a subspace .E of finite codimension in X for which e lE)-' exists and is continuous.We investigate properties of 7 related to the quantities o(T), Pg), p@) and the property T e .Fa and construct a state diagram, called the essential state diagram, analogous to the Taylor-Halbert model [TH] (see also [G1], [G2] and [G3]); the latter will be referred to as the THG state diagram.A consequence is the following observation: 7 is an .F1- operator if and only if its adjoint ?' is a g--operator (Corollary 1.17).Various other state diagrams have appeared in the literature.The reader may consult the survey monograph of V.M. Onieva [O] for further references.Let X , ! , Z, ... denote normed linear spaces.The completion of X will be denoted bV * .The class of linear transformations (henceforth called "operators") ?defined on a linear subspace D(T) of X with range contained in Y is denoted by L(X,Y).The range and null space of T are denoted by ft(?) and nf(7) respectively.The restriction of.T lo a linear subspace M of.X is denoted by TIM; note that TIM :TlM aD(T).The operator 7 is called bounded if 7 is continuous and D(") : X .T is called closedif its graph {@,f *1, o e D(")} is a closed subset of.XxY .Let X7 be the space D(7) normed bV llrllr : llrll+ll?rll .The graph operator Gr of.7 is the operator in L(X7,X) defined by G7r : r (r € X7).W" write G -Gr. Clearly TG is a bounded operator in L(X7,Y).Let E be a linear subspace of.X.Following Pietsch [P] we denote by J § aÅ Qf respectively the natural injection of .E into X and the natural quotient map oiX onto XlE.The adjoint Tt of ? is defined by ?' : (f J §<r)' where the righthand side is the conjugate defined in the usual sense [G3; 50].Note that T' e L(Y',D(T)').We clearly have (/f)' : Q §,, and if .B is closed then (q §)' : Jä:.It is evident that the state diagrams II.3.L4 and II.4.L1 of [G3] are valid in the general case.Given two linear subspaces M, N of X auch that MnN:0 we wrilte M @If for M +N.Let a(T): dim.nr("), 0(T): codim.R(?) and FQ) : codimm ? is called a pq-operator if.o(?) < oo and -B(") is closed, and a g--operator if .R(") is closed ar.dB(T) < oo.If 7 is closed and X and Y are Banach spaces then 7 € .F+ if and only if.T € p+ (Proposition 1.5).The -F'-.,.-operators retain the usual properties associated with