Relatively open operators and the ubiquitous concept

Publicacions Matemàtiques, Vol 38 (1994), 69-79 . RELATIVELY OPEN OPERATOR S AND THE UBIQUITOUS CONCEP T R . W . CROS S A bstract A linear operator T : D(T) c X —+ Y, when X and Y are norme d spaces, is called ubiquitousty open (UO ) if every infinite dimensional subspace M of D(T) contains another such subspace N fo r which T 1 N is open (in the relative sense) . The following proper - ties are shown to be equivalent : (i) T is UO, (ii) T is ubiquitousl y almost open, (iii) no infinite dimensional restriction of T is injective and precompact, (iv) either T is upper semi -Fredholm o r T has finite dimensional range, (v) for each infinite dimensiona l subspace M of D(T), we have dim(TIM) —I (D) + q (T j M) > O . I n case T is closed and X and Y are Banach spaces, T is UO if an d only if TM C T M for every linear subspace M of X . 1 . Introductio n Relatively open operators, i .e . operators for which the image of th e unit ball contains a ball in the range of the operator, have certai n pathologies not shared by their injective components . For instance, the y are unstable under the addition of bounded finite rank operators (cf . [L2, Example 11]), composition, and even under restriction . A new clas s of operators which includes the relatively open operators with finite dimensional kernels, and more generally, F+- operators enjoys a stronge r property which we formulate in terms of the " ubiquitous" concept below . This latter notion was introduced in [C1] where ubiquitous continuity , precompactness and strict singularity were characterised . In the case o f a closed operator between Banach spaces, ubiquitous openess is equivalent to inclusion T M c TM holding for every linear subspace M of th e domain of T (Theorem 3 .11) . Let X and Y be normed spaces and let L (X, Y) denote the class o f linear transformations T : D(T) c X --} Y, where D(T) is the domai n of T . The null space and range of T are denoted by 11T (T) and R(T ) respectively . Write U x = {x EX : i Í x ~ ~ < 1}, the open unit ball of X . 70  R . W . CROS S We call the operator T open (or relatively open) if T(U D(T) ) aU R(T ) for some a > o . If there exists a > 0 such that T(U D(T) ) aU R(T) the n T is called almost open . We adopt the convention that all zero operator s are open and almost open, including the case where D(T) = (0) . The operator T is said to be partially open if there exists a finit e codimensional subspace E of D(T) (or equivalently, of X) for which T 1 E is open . The operator T is called ubiquitously open (UO) if each infinite dimensional subspace M of D(T) contains another such subspace N for whic h T`N is open . Likewise, T is called ubiquitously almost open (UAO) i f it has the corresponding property but with T IN almost open instead o f open . It will be seen that the UO and UAO properties are equivalen t (Theorem 3 .1) . It is clear that in the case of an injective operator the par - tially open and UO properties coincide . An operator P : D(P) c X --~ X is called a projection if P 2 = P . Such an operator is open, and is UO i f and only if either N(P) or R(P) is finite dimensional (Theorem 3 .3) . The operator T is called an F+-operator if there exists a finite codimensional subspace M of D(T) for which T 1M has a continuous inverse . For examples, properties and applications of F+- operators see [C2], [C4] , [C5], [C6], [C7], [C8], [C9], [do], [C11], [CL2], and [L2], [L3] . Theorem 3 .4 states that T is UO if and only if either T is an F+ -operator o r R(T) is finite dimensional . We note some properties enjoyed by the UO operators (Corollary 3 .7) : The product of two UO operators is UO whenever the product is defined . In particular, ifT is UO then so is any restriction of T . If T is UO an d F is any finite rank operator then T + F is UO . The following terminology and notation will be used . Continuou s everywhere defined operators are referred to as bounded . We writ e a(T) -= dim N(T) , and O(T) = dim YI .R(T) . If E is a linear subspace of X then the operator JE (or simply J E ) is the canonical injective map of E into X . Thus JE E 4 E, X) . We define the adjoint of T by T' -- (TJ jj(T) ) ' where the right hand side is defined a s in [G, II . 2 . 2] . The minimum modulus 7(T) of T is defined [K] b y 7(T) = sup {y : { 1 Tx ii 7d(x, N(T)) for all x E D(T)} . Clearly T is open if and only if 7(T) > o . Corollary 3 .6 exhibits a uniform boundedness property enjoyed by the minimum modulus . Let I(X) denote the collection of infinite dimensional subspaces o f X . The quantities q and r (cf . [[C2]) are defined as follows . If RELATIVELY OPEN OPERATORS  7 1 dim D(T) < oo then 0(T) = r(T) = 0 . Otherwis e 0(T) = sup  inf ITN I MEI(D(T)) NEZ(M ) T(T) =  sup  inf UTmI I MEZ(D(T)) T~ EM Il m i l The operator T is called strictly singular ([D], [C2], [C4]) if there i s no M E I(D(T)) such that T~M has a continuous inverse (this is a generalisation of Kato ' s definition [K]). The operator T is strictly singula r if and only if 0(T) = 0, if and only if T(T) = 0 [C2, 3 .4 and 3 .10] , and partially continuous (i .e . continuous on some finite codimensiona l subspace) if and only if 0(T) < oo [C2, 4 .3] . The quantity 1(T) is defined b y «T) a(T) + A(T) . Let X T denote the space D(T) normed by jlxll T = + 11TxII (x E D(T)). The graph operator G T (or simply G) is the canonical injectio n of X T finto X . 2 . Preliminary Proposition s We detail below some known results which are used in the sequel . 2 .1 . Let T be injective and everywhere defined on a Banach space . If T is bounded and if T —1 is continuous on a subspace of finite codimension , then T is an isomorphism {Cl, Theorem 101 . 2 .2 . The operator T is strictly singular if and only if it is ubiquitousl y precompact [C1, Theorem 141, if and only if q (T) = o ([C2, 3 .4] ; se e also [S, 3 .2]) . 2 .3 . Let E be a finite codimensional subspace of D(T) . The n 0(TIE) = 0(T) and T(TIE) = T(T) [C2, 2 .1] . 2 .4 . The operator T is an F+- operator if and only if no infinite dimensional restriction of T is precompact [C2, 2 .2] . 2 .5 . The product of two F+ -operators is an F+ - operator whenever th e product is defined [C2, 2 .13] . 2 .6 . The operator i is open if and only ifTG T is open (see e .g . [L1 , 2 .1]) . 2 .7 . The operator T is almost open if and only if T' is open (see e .g . [MS, 2 .1]) . 72  R . W . CROS S 3 . Ubiquitously open operator s 3 .1 . Theorem . The following statements are equivalent : (i) T is ubiquitously open . (ii) T is ubiquitously almost open . (iii) For each M E I(D(T)) there exists N E Z(M) such tha t 1(TIN) > 0 . (iv) (TIM) > 0 for each M E I(D(T)) . (v) For each linear subspace E of D(T) such that E fl N(T) is finit e dimensional, T1E is an F+-operator . (vi) No infinite dimensional injective restriction of T is a precompac t operator . (vii) Each infinite dimensional restriction of T is partially open . (viii) Some (every) finite codimensional restriction of T is ubiquitousl y open . Proof .• We proceed through the following chain of implications : (i) ~ (iv) ~ (iii) ~ (i), (i) ~ (ii) ~ (v) ~ (vi) ~ (i), (i) ~ (vii) ~ (i), (i ) (viii)  (i) . (i) ~ (iv) : Assume (i) and let 1(TIM) = 0 for some M E I(D(T)) . Then T~M is both injective and strictly singular by 2 .2 and consequentl y for each N E Z(M) we have - y(T I N) = 0, contradicting (i) . (iv) ~ (iii) : Immediate . (iii) ~ (i) : Suppose T is nat UO . Then there exists M E Z( D (T) } such that e y(TEN) = Q for N E Z(M) . Now M n N(T) cannot hav e finite codimension in M (for then TIM would have finite rank, givin g -y (T ~ M) > 0) . Hence by passing to an infinite dimensional subspace of M if necessary, we may suppose that TIM is injective . Then a(T IM) = D and T 1 M is strictly singular, Le . q (T ~ M) = 0 by 2 .2 . Hence 1( T 1M) = D . But then far N E I(M) we have (TIN) = o, contradicting (iii) . (i) ~ (u) : Immediate . (u) ~ ( y ) : Assume (u) . If D(T) is finite dimensional there is nothin g to prove. Accordingly let D(T) be infinite dimensional and let E n N(T ) be finite dimensional where E E Z(D (T)} . Suppose that T E F+ . Writ e S = TJ E ; thus S E L(E, Y) and dim N(S) C oo . Since S « F+ ther e exists M E I(E) such that SIM is a precompact operator by 2 .4 . B y assumption there exists N E I(M) with SIN relatively almost open . Then the adjoint (S 1N} ' is relatively open by 2 .7 and also compact (see RELATIVELY OPEN OPERATORS  73 e .g . [G, III .1 .11]) . Therefore (S i N) ' = (SJN) 'has finite dimensiona l range . But R((SJ N ) ' ) L = N(SJ N ) is finite dimensional . Therefor e dimR((SJ N ) ' ) = dimR((SJ N ) ' ) L = dimN((SJ N ) 1 = dim(E/N(SJ N ))' = dimE/N(SJ N ) = oo , a contradiction . Hence (ii) ~ (v) . (v) ~ (vi) : Assume (v) and letTEM be injective where M E I(D(T)) . Then T is an F+-operator . Therefore T~M is not precompact . (vi) ~ (i) : Suppose T is not UO. Then by the equivalence (i) ~ (iv) aboye, there exists M ~ I(D(T)) such that TEM is injective an d strictly singular . In that case there exists N ~ Z(M) for which T 1 N i s precompact by 2 .2, contradicting (vi) . (i) ~ (vii) : Assume (i) . Let M be any linear complement of N(T) i n D(T) ; thus D(T) = M + N(T) where M f1 N(T) = 0 . Then T M is a n F+-operator (by (i) ~ (v)) . Hence then exist linear subspaces E and F of M such that M = E + F, where F is finite dimensional and (TSE) — 1 is continuous . It only remains to verify that TSE+N(T) is relatively open . We hav e y(T IE + N(T)) = sup{ry : IIT(e + )II > ryd(e + n, N(T) ) foralleEE,nEN(T) } = sup{y :  7d(e, N(T)) for all e E E } = sup{ry  > y llell for all e E E } = ry(T I E) > 0 as required . (vii) ~ (i) : Suppose that T is not UO . Then there exists N E Z( D (T) } for which Tm is injective and precompact (by (vi) ~ (i) above} clearl y contradicting (vii) . (i) ~ (viii) : Both statements are immediate . (viii) ~ (i) : It is clearly sufficient to prove the first statement . W e may clearly suppose that D(T) is infinite dimensional . Let E be a finit e codimensional subspace of D(T) for which T E is UO . Let M E I(D(T)) . Then 0(T1M) = 0(TIE f1 M) by 2 .3 and a(T I M) > a(T I E n M) . Consequently 1(TIM) > e (T I E fl M) . Hence T is UO is by the equivalenc e (i) ~ (iv) . ■ 74  R . W . CROS S 3 .2 . Corollary . Let /3(T) < oo . Then the following statements ar e equivalent : (i) T is almost open . (u) T' is ubiquitously open . (iii) T' is a 0+-operator . Proof : We have ,3( T ) = ct ( T' ) c cc . Now T is almost open if and onl y if T ' is open (2 .7) if and only if T ' is a 0 +- operator (see e .g . [G, IV .1 .61) , if and only if T ' is ubiquitously open (by Theorem 3 .1) . ■ 3 .3 . Theorem . Let P be a linear projection . Then P has an infinit e dimensional injective precompact restriction if and only if both N(P) an d R(P) are infinite dimensional . Proof : Let M be an infinite dimensional subspace of D(P) for whic h P IM is injective and precompact . If dim N(P) < oo then R(P) ha s finite codimension in D(P) and hence IVI n R (P) is an infinite dimensiona l subspace upon which P is both an isomorphism and precompact, which i s impossible . Therefore dim N(P) = oo . On the other hand if dim R(P) < co then P does not have an infinite dimensional injective restriction, a contradiction . Hence necessity follows . Conversely suppose that both N(P) and R(P) are infinite dimensional . Choose a linearly independent sequence (a a ) in R(P) such tha t 0 . Next construct a sequence (b u ) in N(P) such that llb n II - = 1 and llbn - - b k II > 1 (n ~ k) (for example, by the Gramm-Schmidt construction) . Write x n = an + b n and E = sp{x n } . Then PIE is injective ; indeed e = EcYzxi, Pe = 9 ~ E cx i Px i = Ecxiai = o e — 0 by th e linear independence of {a} . Furthermore PX n = an —> o and (x) ha s no Cauchy subsequence since II x n — x k II > Iib n — b k 11 — ii an - -- a k II > eventually . Therefore PIE is not an E+— operator by [C4, 21] and henc e has an infinite dimensional precompact restriction [G, 111 .1 .9] . ■ 3 .4 . Theorem . The following are equivalent : (i) T is ubiquitously open . (ii) Either T is an F+-operator or R(T) is finite dimensional . Proo f- For simplicity of notation let D(T) = X . Assume (i) an d suppose in the first instance that T is continuous . Write X = N(T) + E where N(T) n E = {9}, S = (TIE)' and P = ST . Then P is a projection with N(P) = N(T) and R(P) = E . Let M E I(X) . By RELATIVELY OP EN OPERATOR S hypothesis there exists N E Z(M) for which -y(T I N) > O . We hav e ( P I N) = (ST I N) —  n + II STn M  N(T N il —  ii n inf  + S N II T N = 1 y ( S )7( T ~ N ) > 0 since y(S) > 0 by the continuity of T . Therefore P is UO . By Theorem 3 .3 either dim R(T) = dim E < oo or dim N(T) < oo . I n the latter case T E F+ . Hence (i) ~ (ii) in the case when T is continuous . Passing now to the general case when T is arbitrary, we note that fo r any subspace M of X , r y(T PI) > 0 ~-y(TGIG -1 M) = r Y(TM o G T 1 m) >o by 2 .6 and hence T is U0 if and only if TG is UO . Since TG is continuous , it follows from what has been proved that either R(T) = R(TG) is finit e dimensional or T G E F+ . But clearly TG E F+ T E F+ . Therefor e (i) ~ (u) . Conversely assume (u) . If R(T) is finite dimensional then clearly T is UO . On the other hand if T is an F+ -operator then so are all it s restrictions and hence by (vii) of Theorem 3 .1, T is UO . ■ 3 .5 . Corollary . Let X and Y be Banach spaces and T a closed operator with infinite dimensional range. Then the following are equivalent : (i) T is a 0+-operator . (ii) T is ubiquitously open . Then next corollary is a uniform boundedness principie for the minimum modulus function e y (T) . 3 .6 . Corollary . The following statements are equivalent for an arbitrary operator T : D(T) c X —> Y with infinite dimensional domai n D(T) : (i) For each M E Z(D(T)) we have sup ry(T1N) > 0 . NEI(M ) (ii) inf  sup 7( T I N ) > O . MEI(D(T)) NEI(M ) Proof : Assume (i) . Then T is UO . Hence either dim R(T) < o0 o r T E F+ by Theorem 3 .4 . In the former case ry(TiN) = oo for eac h N E I(N(T)) and so (ii) holds . In the latter case we hav e inf  sup ry(T1 N) > infsup ñn Ñ 11 n  ME Z(D(T) ) inf  T(TM) . MEZ(D(T)) 1yEZ(M)  M N 75 76  R . W . CROS S Let E be a finite codimensional subspace of D(T) for which (TIE)- ' exists and is continuous . Then M n E has finite codimension in M fo r each M E Z(D (T)} and then r(T 1 .lVl ) = -r (T n E) by 2 .3 . Therefor e inf  f MEI(D(T)) r(TM) ME (E) T (T I M ) — O (( T E)-i - 1 J But (TIE) —1 is continuous and so 0((TIE) —1 ) < oo . Hence (ii) follows . ■ 3 .7 . Corollary . (a) The sum of a ubiquitously open operator and a finite rank operato r is ubiquitously open . (b) The product of two ubiquitously open operators is ubiquitousl y open . Proof : (a) Immediate from the equivalence (i)  (viii) of Theorem 3 .1 . (b) Combine Theorem 3 .4 with 2 .5 . ■ ~ 3 .8 . Corollary . If T is ubiquitously open then so is T . According to Corollary 3 .8 the continuous linear maps between coincomparable Banach spaces [GO], except those of finite rank, are all no t UO. The situation is similar when the two spaces are totally incomparable Banach spaces (cf . [R]) ; indeed such maps are strictly singular . ~ 3 .9 . Proposition . L et T be partially open . Then T is partially open . Proof : Let E be a finite codimensional subspace of D(T) for which T E is open . Write Q = QÑ (T) . We hav e 7'Q e 7(7)(2E ) R F% ) = éÉÉ Q e > e in f EE I) e + I N ( T E) (I - -y(T E ) > 0 . „  ~ Since QE is finite codimensional in Q (D (T )) = D (T), T is partiall y open . ■ 3 .10 . Proposition . Let X and Y be Banach spaces and T a close d operator . Then T is partially open if and only if it is open . RELATIVELY OPEN OPERATORS  77 Proa : Recall that T if RO if and only if TG T is RO (2 .6) . Henc e also T is PRO if and only if TG T is PRO . Furthermore X T is a Banac h space . Consequently we may suppose that T is bounded . Let T b e , partially open . Then T is partially open by Proposition 3 .9 . Hence T i s an isomorphism by 2 .1 . Therefore T is open . The converse statement i s trivial . ■ 3 .11 . Theorem . Let X and Y be Banach spaces and T : D(T) C X—> Y a closed operator . Then the following statements are equivalent : (i) T is either a 0 +- operator or a bounded finite rank operator . (u) Por each linear subspace M o f D(T) we have TM C TM . Proof .• Assume (i) . Then T is U0 by Theorem 3 .4 . Let M be a n arbitrary subspace of D (T ) . By Theorem 3 .1 T 1M is partially open an d hence open by Proposition 3 .10 . Therefore T (I1rI) is closed by [G, IV .1 .6 ] . Hence (i)  (u) . Conversely, assume (u) . Let R(T) be infinite dimensional . Suppos e M E Z( D (T)} where M n 1V (T) = (0) . If T IIUI is compact, then by th e density invariant result ro (T I 11r1 ) = r o (T IM) [Cli, 3 .7], TIM is als o compact, while T(M) contains the infinite dimensional Banach spac e TM, which is impossible . Hence T is UC by Theorem 3 .1 and consequently T is a 0 + -operator by Corollary 3 .5 . Since closed finite ran k operators are bounded, the conclusion (i) now follows . ■ Reference s [CI] R . W . CROSS, Some continuity properties of linear transforma - tions in normed spaces, Glasgow Math . J . 30 (1988), 243-247 . [C2] R . W . CROSS, Properties of some norm related functions of unbounded linear operators, Math . Z . 199 (1988), 285-302 . [C3] R . W . CROSS, Unbounded strictly singular operators, Proc . Netherlands Acad . Sc . Series A - Indag . Math . 91, no . 3 (1988) , 245-248 . [C4] R . W . CROSS, Linear operators of upper semi ---Fredholm type i n normed spaces, Portugal . Math . 47, no .1 (1990), 61-79 . [C5] R . W . CROSS, The essential state diagram of a linear operator , Ann . Acad . Sci . Fenn . Ser . A .I . Math . 15 (1990), 283-291 . [C6] R . W . CROSS, On the perturbation of unbounded linear operator s with topologically complemented ranges, J . Funct . 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