Embry-Wardrop, Mary R.

English

Embry-Wardrop Mary R.

University of Szeged

A generalization of the Halmos—Bram criterion for subnormality By M A R Y R. EMBRY in Charlotte (North Carolina, USA) Introduction. In [3] and [1] HALMOS and BRAM show that a continuous linear oper-n ator A on a complex Hilbert space Xis subnormal if and only if ^ (A'xj, AJxt) = 0 l,J = 0 for all finite collections x0, ..., x,\ of X. In Section 1 we generalize this criterion n by showing that is subnormal if and only if ^ (AI+JXJ, AI+JXI) S 0 for all i,j=0 finite subcollections x0,...,xn of X. As an application of this criterion we show in Section 2 that an operator A is the restriction of a normal partial isometry to an invariant subspace if and only if A—A*A2 and |[/ij[ s 1. In Section 3 we show, using our new criterion for subnormality, that an operator A is subnormal if and only if {A*"A"}^L0 is a Hausdorff moment sequence. Throughout the paper X is a complex Hilbert space with inner product ( , ) and norm || ||. If B is a continuous linear operator on X, then B* is the adjoint of B. B is normal if BB*=B*B, quasi-normal if B(B*B)=(B*B)B, an isometry if B*B=I and a partial isometry if (B*B) 2 = B*B. An operator A is subnormal if it is the restric-tion of a normal operator B to an invariant subspace of B and hyponormal if AA*'S SA*A. A sequence {C„}~ of operators on A'is a Hausdorff moment sequence if there b exists a positive operator measure <p on some interval [a, b] such that Cn = f t " dcp ' for each nonnegative integer n. " 1. A criterion for subnormality. The Halmos—Bram criterion that an operator n A on Xbe subnormal is that ^ (A'xj, / i J x , ) = 0 for all finite collections x 0 , . . . , x„ ' of X. We generalize this as follows: T h e o r e m 1. An operator A on a complex Hilbert space X is subnormal if and only if A satisfies (S,) 2 (A'+Jxj,A'+Jx,)s0 i,j = o for each finite collection x0, ..., xn of X. 62 M. R. Embry P r o o f . The proof of the necessity of the condition is easy. Note that (Sx) is the special case of the Halmos—Bram condition which we obtain by choosing Xi = Aix'l for 1=0, To prove the sufficiency of the condition we imitate the techniques of Halmos and Bram and prove that if A satisfies condition (Si) , then A is the restriction of a quasi-normal operator to an invariant subspace. This will complete our proof, since every quasi-normal operator is subnormal ([4, problem 154]). Assume now that A satisfies condition (Si) . The first step in the proof will be to show that A also satisfies (S2) ¿(A,+J+1Xj,A,+J+1xi)^\\A\\2 2 (^'+Jxj,A'+Jxd. i,j = o. i,j=0 To obtain this result we outline.a modification of BRAM'S proof in [1, Theorem 1, p. 76]. , . Let e > 0 and let A^AKWAW+e). A t also satisfies condition (5 , ) . Let Y=l2(X). Define C o n 7 by (Q>) f= 2AV+JA+Jyj- An argument similar to that i= o used by Bram shows that C is a well-defined, bounded operator on Y and that C ^ O on Y. Now define B on Y by (By)i = Ai yt. A computation almost identical to that used by Bram shows that \\B*CBy\\^\\Cy\\ for all y in Y and hence by [5, p. 426] that B*CBsC since ||5|| = | U , | i < l . It now follows that if x 0 , ...,x„ are elements of X, then •2 (Ai+J+1xJ,A'+i+1xd*(\\Ayi+*)2 2 (A'^Xj^'^Xi). 1,7 = 0 i, J=0 Since E was an arbitrary positive number, condition (S2) is satisfied. The second step in the proof of the theorem is the construction of a quasi-normal extension of A. The following modification of HALMOS' proof in [3] will give us this result. Let X be the set of all sequences in X such that x ( = 0 for 0 and x ^ - 0 for at most a finite number of /. On X define (x,y)= 2(Ai+Jxj,Ai+jyd. i.j Let F b e the set of equivalence classes obtained by identifying x with 0 if (Jc, jc) = 0. Then since A satisfies condition (S1,), ( , ) is an inner product on Y. Define D on Criterion for subnormality 63 X by (Dx)j=AXi. Using the fact that A also satisfies condition (S2), we have (.Dx, Dx) — 2 ¿ » ' " x j ^ M 2 2 (Ai+Jxi, Ai+'xj) = \\A\\2(x, x). i,j i,j It now follows that D may be considered to be a cont inuous linear operator on f . Define E on X by (Ex)i=xi-1 and note tha t DE=ED on Fur thermore on X we have the relation (Dx, Dy) = Z(Ai+J+1Xi, A'+j+1Xj)= 2(Al+JXi, Ai+JxJ_1) = (x, Ey). U i.J Thus on the completion of % the extensions of D and E satisfy the equat ion E=D*D. However, we have already observed that D commutes with E. Therefore the extension of D to the completion of Y is a quasi-normal extension of A and the proof of the theorem is complete. In [6, Theorem 7, p. 73] MAC NERNEY shows that a sequence {C„}"=0 of Her-mitian operators on J is a Hausdorff sequence for the interval [a, b] if and only if n n n a 2 (xi> Ci+jxj) — 2 (xi,Ci+J+1Xj)^b 2 (Xi, Ci+jXj) i,j= o i,j-0 i,j=0 for each finite collection x0, ..., xn in X. Using this result and Theorem 1, we readily obtain the following: C o r o l l a r y . An operator A on X is subnormal if and only i f , {A*"An}^=0 is a Hausdorff moment sequence. We note that if A is subnormal , then A is quasi-normal if and only if A*"A" = = f t"d(p where cp is a spectral measure (that is, tp is a projection-valued operator measure). The proof of this assertion is simple. If A is quasi-normal, then A*nA" = =(A*A)n fo r nSO and thus A*"A" —• f t"d(p where <p is the spectral resolution of A*A. Conversely, if A*"A" = f t"d(p and <p is projection-valued, then A*"A" = = (A*A)" for /2^0 . Fur thermore , by the last corollary A is subnormal and hence hyponormal . However if A is hyponormal and A*2A2 = (A*A)2, then (A*A — AA*)A = 0, proving that A is quasi-normal. 2. The operator equation A =A*A2. Consider the weighted shift A on I2 defined by A(x0, x1; ...) = (0, 2x0, xl,x1, ...). A simple computat ion shows tha t A=A*A2. However, since the weights of A are not monotone increasing, A is not hypo-normal [4, p. 160] and consequently not subnormal . Thus not every opera tor satisfying the equat ion A =A*A2 is subnormal . The additional hypothesis needed to force A to be subnormal is tha t | | / i | | S l . We are now able to completely cha-racterize operators satisfying these two conditions. % 64 M. R. Embry T h e o r e m 2. Let A be an operator on a complex Hilbert space X. A is sub-normal and the minimal normal extension of A is a partial isometry if and only if Mil S i and A = A*A2. P r o o f . Assume first that B is a normal partial isometry on a Hilbert space Y, containing X, and that B=A on X. Since every partial isometry has norm s i , we have | | ,4 | |S | | .B | |Sl . Let P be the projection of Y onto X. Then for each x in X we have A*A2x—PB*B2x=PBB*Bx (since B is normal) = (since B is a partial i some t ry )—Bx=Ax and consequently, A=A*A2. Now assume that A=A*A2 and M | | S l . A simple inductive argument shows that A*kAk=-A*A for each integer / c £ 1. Therefore if x 0 , . . . , x„ are elements of X, 2 (Ai+Jxj,Ai+Jx)= 2 {Axj,Axl) + \\x0\\2-\\Ax()\\2 = i,j= 0 i,j = 0 2 Axi 1 = 0 + ll*o II P*ol l s 0 since Ml! s i . By Theorem 1 we know that A is subnormal. Let B\Y-*Y be the minimal normal extension of A. It remains to show that B is a partial isometry. Let P be the projec-tion of Y onto X. Then for x in X, \\PB*B2x\\ = \\A*A2x\\ =\\Ax\\ =\\A3x\\ (since A*3A*=A*A) = lBzx\\ = \\B*B2x\\. Therefore B*B2x£X for all x and consequently B*B2 = B on X. Since B is the minimal normal extension of A, the set of vectors | ^ is dense in Y and consequently B*B2=B on a dense subset of Y. This is sufficient to imply that B is a partial isometry. The proof is complete. The assertion in Theorem 2 parallels the assertion that an operator A is an isornetry if and only if A is subnormal and the minimal normal extension of A is unitary. References [1 ] J . B R A M , Subnormal operators, Duke Math. J., 2 2 ( 1 9 5 5 ) , 7 5 — 9 4 . [ 2 ] R . G . D O U G L A S , On.majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), 413—415. [3] P. R. HALMOS, Normal dilations and extensions of operators, Summa Brasil. Math., 2 (1950), 125—134. [4 ] P. R . HALMOS, A Hilbert space problem book, Van Nostrand Co. (Princeton, 1 9 6 7 ) . [5] E . H E I N Z , Beiträge zur Störungtheorie der Spektralzerlegung, Math. Ann., 1 2 3 (1951), 415—438. [6] J. S. M A C N E R N E Y , Hermitian moment sequences, Trans. Amer. Math. Soc., 103 (1962), 45—81. UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE (Received October 25, 1971) 

A generalization of the Halmos-Bram criterion for subnormality

http://acta.bibl.u-szeged.hu/14430/1/math_035_061-064.pdf

A generalization of the Halmos-Bram criterion for subnormality

Abstract

Similar works

Full text

Available Versions

University of Szeged