Null spaces and ranges of polynomials of operators

Publicacions Matemátiques, Vol 32 (1988), 167-170 . Abstract NULL SPACES AND RANGES OF POLYNOMIALS OF OPERATORS MANUEL GONZÁLEZ We give an elementary proof of the fact that given two polynomials P, Q without common zeros and a linear operator A, the operators P(A) and Q(A) verify some properties equivalent to the pair (P(A), Q(A)) being non-singular in the sense of J .L . Taylor . From these properties we derive expressions for the range and null space of P(A) and spectral mapping theorems for polynomials of continuous (or closed) operators in Banach spaces . Let X be a complex linear space, A E L(X) a linear map on X . If A,,u E E C,,\ :~ y, then (*)  (A - A)x = (A - p)x + (y - A)x for each  x E X From this simple equality it follows a big amount of information about the relations between the null spaces and ranges of S= A - A and T = y - A . In fact we have : (a)  N(S) n N(T) _ {0} (b)  R(S) + R(T) = X (c)  T[N(S)] = N(S) ; S[N(T)] = N(T) (d)  R(ST) = R(S) n R(T) (e)  N(ST) = N(S) ® N(T) (a), (b) and (c) follow immediatly from (*), and as ST = TS R(ST) C R(S) n R(T) ; but if z E R(S) n R(T), then z = (A - A)x = (,u - A)y . Using (*) we obtain y = (y - A)_ 1 (A - A)(x - y) . Hence z = (y - A)(A - A)[(p - A) -1 (x - y)] E R(ST) . Finally by (c) we have T -1 [N(S)] = N(S) + N(T) . Then taking account of (a) we conclude N(ST) = N(S) ® N(T) . we have 168  M . GONZÁLEZ In this note we shall prove that given P(z) and Q(z) polynomials without common zeros, the operators S = P(A) and T = Q(A) also verify the five properties above . As a corollary we obtain expressions for R[P(A)] and N[P(A)] as functions of the ranges and null spaces of ( .\ - A)', .\ zero of P(z) of order n , as is obtained in [4] with a different proof, and some other interesting consequences related with the spectral mapping theorem for polynomials as given in [1] . We start proving an auxiliary result . Proposition . Let S, T E L(X) such that ST =TS . (i) S and T verify (c) and (d) if and only if given y, z E X such that Sy = Tz, there exists x E X with y = Tx and z = Sx . (ii) If S and T verify (a) and (c), then they verify (e) . Proof . (i) Suppose S and T verify (c) and (d), and Sy = Tz = u . As u E E R(S) n R(T) = R(ST), there exist v such that u = STv = TSv . Then Tv - y E N(S) and Sv - z E N(T) . Using (d) we can choose f E N(S) and gE N(T) such that Tv - y =T f and Sv - z = Sg . Taking x = v - f - g we have Tx = yand Sx = z . Conversely, let S and T verifying the second part of (i) . We have R(ST) C C R(S) nR(T) . Let v E R(S) nR(T) . Then v = Ty = Sz for some y, z . Taking x such that z =Tx we obtain v = STx E R(ST) . Moreover T[N(S)] C N(S) . Let y E N(S) . TO = Sy = 0and then there exist x such that Sx = 0 and Tx = y . Hence T[N(S)] = N(S) . Analogously S[N(T)] = N(T) . (ii) As T[N(S)] = N(S), taking account of (a) we conclude N(ST) _ =T -1 [N(S)] = N(S) ® N(T) . E Definition . Let S, T E L(X) a pair of commuting operators . If S and T verify (a), (b), (c) and (d) (and then also (e)), we shall soy that the pair (S, T) is non-singular . Note that the above definition coincides with the non-singularity in tlie sense of J .L . Taylor [3], used to define a joint spectrum for two commuting operators . We don't follow that way . The following theorem is the central result in this note as it will allow us to pass from an operator to their polinomials and vice versa . Theorem . Let A, B, T E L(X) commuting operators ; S = AB . (S, T) is non-singular if and only if (A, T) and (B, T) are non-singular . Proof .. Suppose (S, T) non-singular . As N(A)+N(B) C N(S) and R(S) C C R(A)fnR(B) we obtain (a) and (b) for (A, T) and (B, T) from these properties for (S, T) . Let y, z E X such that Ty = Az . By (b) there exist u, v E X such that z = Bu +Tv and so Az = ABu+ATv = Su +TAv . Then Su = T(y-Av) . POLYNOMIALS OF OPERATORS  169 Let w such that Tw = u and Sw = y - Av .  Taking x = Bw + . v we have Tx = TBw + Tv = Bu + Tv = z, and Ax = ABw + Av = Sw + Av = y . Using the above proposition we conclude that (A, T) is non-singular . The proof for (B, T) is the same . Suppose now (A, T) and (B, T) non-singular . If xE N(S) fl N(T), then Ax E N(B) n N(T) = {0}, and so xE N(A) fl N(T) . Hence x = 0 . Let xE X . x = Ty + Az for some y, z in X, and z = Tu+ Bv for some u, v in X . Then x =Ty+ATu+ABv =T(y+Au)+Sv . Hence R(S)+R(T) =X . Finallylet Sy = Tz . As ABy = Tz there exist v in X such that Av = zand Tv = By . Now we can take x in X such that Tx = y and Bx = v ; and so Sx = Av = z . Hence we conclude that (S, T) is non-singular . Corollary . Let A E L(X) and P(z) and Q(z) polynomials without common zeros . Then we have (i) (P(A), Q(A)) is non-singular . (ii) If P(z) = (A 1 - z)" . . . ( , \k - z)rnk with mi integers and Xi : Aj for i  j then R[P(A)] = R[(a, - A)"] n . . . n R[(a k - A)mk] N[P(A)] = N[(-\, - A)"] ® . . . ® N [(a k - A)'k] (iii) If X is a Banach space and A is continuous, then R[P(A)] closed if and only if R[(,Xi - A)'¡] closed ; i = 1, . . . , k . R[P(A)] dense if and only if R[(Aj - A)'", ] dense ; i = 1, . . . , k . Proof .. (i) If P(z) = ('\1 - z) m1 . . . (Ak - z)nk and Q(z) _ (pi - z)"' . . . . . . (h9 - z)"^ then ( ,\ i - A, j j - A) is non-singular for every i and j, and the result follows immediately from the theorem . (ii) If \ i :~ \ j then ((,\i - A)'",, (A j - A)''¡) is non-singular .  The results now are consequence of (d) and (e) taking account of the proposition . (iii) If R[(Aj - A)'¡], i = 1, . . . , k are closed then from (ii) it is evident that R[P(A)] is closed . On the other hand, making B = (A 1 - A)" . . . . . . (Ak-1 - A) Mk- ', by (ii) and (c) we have N(B) C R[( , Xk - A)mk] . Hence if R[P(A)] is closed, then B-1(R[P(A)]) = R[(,\k - A)" : ] + N(B) = R[(,Xk - -A)" . k] is closed ; and as ( ,\ i - A)- are commuting operators, we prove in the same form that they all have closed range . The final result follows from (ii) taking account that a continuous operator B E L(X) has dense range if and only if the conjugate B' is injective [2] . Remarks (1) The above theorem, of course, can be obtained from the spectral mapping theorem for the Taylor spectrum associated to several commuting operators [3], but our version of the proof totally avoids Liouville's theorem in several complex variables, and in this sense is elementary . (2) It is evident that the above corollary admits some generalizations : 17 0  M . GONZÁLEZ Parts (i) and (ii) can be extended to linear operators with domain D(A) :~ Jr' such that there exist aE C with aA bijective . It suffices to consider factors of the form (A - A)'(a - A) -k , as the operator P(A) has the same range and null space that P(A)(a - A)- ", where n is the grade of the polynomial P(z) . Analogously part (iii) can be extended to closed operators with nonvoid resolvent set . Note that in this case, by the closed graph theorem, (,\ - A)'(ce - A) -k is continuous . References 1 .  H . A . GINDLER, A spectral mapping theoremfor polynomials, Amer . Math . Monthly 72 (1965), 528-530 . 2 .  A . E . TAYLOR, D . C . LAY, "Introduction to functional analysis," 2nd . ed ., Wiley, 1980 . 3 .  J . L . TAYLOR, The analytic functional calculus for several commuting operators, Acta Math . 125 (1970), 1-38 . 4 .  T . YAMAMOTO, A note on the spectral mapping theorem, Siam J . Math . Anal . 2 (1971), 49-51 . Departamento de Matemáticas Facultad de Ciencias 39005 Santander, SPAIN . Rebut el 15 de Febrer de 1988