ON THE UNILATERAL SHIFT AND COMMUTING CONTRACTIONS

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1 ON THE UNILATERAL SHIFT AND COMMUTING CONTRACTIONS JUSTUS K. MILE Kiriri Women s University of Science & Technology, P. O. Box Nairobi, Kenya Abstract In this paper, we discuss the necessary and sufficient conditions for an operator in a Hilbert space to be unitarily equivalent to a unilateral shift. Consequently, we discuss the problem when a commutative sequence of operators is similar to a commutative sequence of contractions, which has a unitary dilation. Introduction In this problem, X will represent the set 1,2,,;h ; Z is the set of all integers and #(%) is the abelian group of all functions ):+ - with definite supports. Let H, K be complex Hilbert spaces with the inner product.,. and norm % = %,7 8/: % <. B(<) stands for the Banach algebra of all bounded linear operators in H. An operator U in H is a unilateral shift if and only if U is an isometry satisfying, n H =0. For a non-empty set B, let F G : (H) denote the Hilbert space of all systems I = (I ) K of vectors I < such that M I N N : < + and the inner product defined by I,R = M I,S,TUV I = (I ),R = (S ) F G : (H). The space F G : (# X (%)) will be denoted by < Y. 109

2 We define the operator [ \ in < Y (] %) by I `ab (^\ I) = _ 0, \ N Preposition 2. The operator [ \ is pure isometry. The multiplicity of [ \ equals the dimension of < \ = I < Y I = 0 and it does not depend on m X. The operators U m commute with each other. A commutative sequence R = (h \ ) \ i of operators in H is similar to a commutative sequence j = (j \ ) of operators in K, if there is a homeomorphism V : H K such that We then write R ~ Q. Proposition 3. If R~j,ph lh \ = j \ l for all m X. l <,h > = < l, >, h =<,j > l TUV rsh t X (%), (1) h ~ j, where Y= 1 V * (2) Theorem 4. (Main Result). Suppose h = (h \ ) \ Y is a commutative sequence of operators in a Hilbert space H. If R <, then R is similar to a common part of the sequence [. Proof. We define ) < < Y by () I) =,h I, for all I <, # X (%) The following inequalities hold N I :N = I,I ) I,)I ) N I :N N h : N I :N, I <. (3) 110

3 Suppose M =Y H. It follows that from (3) that M is a closed subspace of H x. Also, by Proposition 2, we have ) h \ I }( + \,h I) ~ = [ \ ) I, I < and thus ) h \ = [ \ ). (4) Thus this implies that each operator [ ] has M invariant. Define l < S, ph li = )I TUV I <. We therefore have from (3) and (4) that V is a linear homeomorphism such that l h \ = [ \ ] N, ] %. We know that a commutative sequence j = (j \ ) \ Y of operators in the Hilbert space H has a unitary dilation, if there is a Hilbert space S < and a commutative sequence l = (l \ ) \ Y of unitary operators in K such that,j =,l < N, # X (%), where P is the orthogonal projection of K onto H. Corollary 5. Suppose that h = (h \ ) \ Y is the commutative sequence of operators in a Hilbert space H. If h < +, then there is a Hilbert space K and a commutative sequence j = (j \ ) \ Y of contractions in K such that h~j r j has unitary dilation. Proof. Now h = h <, it follows by Theorem 4 that there exists a subspace M of H such that each operator [ \ leaves M invariant and h ~j = (j \ ) \ Y, here j \ = [ \ proposition 3, we have h~j = j. The sequence j = (j \ ) \ Y consists of mutually commuting contractions ] Ṇ Therefore by j \ = ([ ] ] N ) = [ ] ] Ṇ We have <,j > = Œ ]: (]) 0 N([ \ ] N ) (]) = Œ<,[ > ] N = <,[ > ] N Thus, <,j > = <,[ > ] N, # X (%). We define the unitary operator l \ F 2 < }#(%)~ 7 (l \ I) `ab, I F 2 < }#(<)~. Identifying < Y with the subspace ŒI F 2 < }#(<)~:I = 0 TUV rsh # X (%) of F 2 < }#(<)~,ph l \ < Y < Y,r l \ I = [ \ I, I < Y. 111

4 Preposition 5 An operator U on H is unitarily equivalent to a unilateral shift if and only if U is an isometry and dim V(H). [ (<) = 0 Proof. Let U be an isometry satisfying the condition [ (<) = 0 Let S = ([(H)). Clearly S [(<). p S = [ \ (S ). > 0. Now S [(<), > 0;r s S [(<), U S S. Also for all F 0 SF = [ ž (S ) [ ž (S ) = [ ž }[ (S )~ = [ žx (S ) = S žx, F 0,r 1. So, [ (S ) [(S Ÿ ). Again applying [,[ : (S ) [ (S Ÿ ), ps. Conclusion S = 0,1,2, is an orthogonal family of subspace of H. Since S = ([(H)), so by the projection theorem < = [ (<) S. (5) Since U is an isometry, [ (<) = [ : (<) [(S ) = [ : (<) S 8 (6) Substituting for U (H) in (5) from (6), we get < = [ : (<) S 8 S. (7) So, [(<) = [ (<) [(S 8 ) [(S ) = [ (<) [ : (S ) [ (S ) = [ (<) S : S 8 (8) 112

5 Substituting in (3), we get < = [ (<) S : S 8 S. Continuing in this manner, we obtain < = [ (<) S `8 S `:..S (9) Let - < ^sh phrp - S, 0. <s [ (<), 0. But [ (<) = 0,(I ). - = 0 s rv F7. < = M S = S [(S ) [ : S. Consider the unilateral shift [ X : F : (S ) F : (S ), with multiplicity dim([(<)) = dim S. : Define :F X (S ) < 7 (%,% 8, % :, ) = % +[% 8 + [ : % : +. The RHS represents a vector in H for % +[% 8 + [ : % : + and S are orthogonal and % ª are in S ; % +[% 8 + [ : % : +.. Ṇ :N = % : + N [% 8 : N [% : N : + (Pythagoras theorem) N= % : + N % 8 : + N [% : N : + N= % : + N % 8 : + N % : N : + < N = % N :, where % = (%, % 8, % :, ). We have also shown that N % N :N = % N :. W is clearly linear and thus W is an isometry. W is also onto. Finally, we observe that [ X = [ TUV [ X (%, % 8, % :, ) = (0,%, % 8, % :, ) = ([% + [ : % : + ) and 113

6 [ (%,% 8, % :, ) = [(% +[% 8 + [ : % : + ) = [% +[ : % 8 +[ % : + [ = [ X, i.e., [ = [ X `8 U is unitarily equivalent to [ X. The converse is obvious. For if U is unitarily equivalent to a unilateral shift (of multiplicity n), then U itself must be a unilateral shift of the same multiplicity. Put [(<) = S, hvdims = Let [~l, where V is a unilateral shift. Then U=W* VW for a unitary W. Hence for all % <, [% = l % = (l )% = l % = % = % (Since W is unitary) [ is isometric. < = [ < S `8 S, Hence, we can write < = [ < S `8 S, S = ([(<)) r S = [ S, 0. Now [ < [ X8 <,. Hence As, hr [ < S `8 S,. ª < = [ < M It is thus evident that U is a unilateral shift and ([<) = S, then Hence S < = S + S 8 + = M S. [ (<) = 0. (10) 114

7 References [1] J. A. Ball, Rota s theorem for general functional Hilbert spaces, Proc. Amer. Math. Soc. 64(1) (1977), [2] Peter R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer-Verlag, New York, [3] S. K. Berbarian, Introduction to Hilbert Spaces, Chelsea Publishing Company, New York, [4] Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York, London, [5] Béla Sz.-Nagy and Ciprian Foiaş, Harmonic Analysis of Operators on Hilbert Space, Akadémiai Kiadó, Budapest,