Wandering subspaces of the Bergman space and the Dirichlet space over polydisc

Transcription

1 isibang/ms/2013/14 June 4th, statmath/eprints Wandering subspaces of the Bergman space and the Dirichlet space over polydisc A. Chattopadhyay, B. Krishna Das, Jaydeb Sarkar and S. Sarkar Indian Statistical Institute, Bangalore Centre 8th Mile Mysore Road, Bangalore, India

2

3 WANDERING SUBSPACES OF THE BERGMAN SPACE AND THE DIRICHLET SPACE OVER POLYDISC A. CHATTOPADHYAY, B. KRISHNA DAS, JAYDEB SARKAR, AND S. SARKAR Abstract. Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc D n (with n 2) is investigated. We show that for any non-empty subset α = {α 1,..., α k } of {1,..., n} and doubly commuting invariant subspace S of the Bergman space or the Dirichlet space over D n, the tuple consists of restrictions of coordinate multiplication operators M α S := (M zα1 S,..., M zαk S ) always possesses wandering subspace of the form k (S z αi S). 1. Introduction A closed subspace W of a Hilbert space H is said to be wandering subspace (following Halmos [Hal]) for an n-tuple of commuting bounded linear operators T = (T 1,..., T n ) (n 1) if for all (l 1,..., l n ) N n \ {(0,..., 0)} and W T l 1 1 T l 2 2 T l n n W H = span{t l 1 1 T l 2 2 T l n n h : h W, l 1,..., l n N}. In this case, the tuple T is said to have the wandering subspace property. The main purpose of this paper is to investigate the following question: Question: Let (T 1,..., T n ) be a commuting n-tuple of bounded linear operators on a Hilbert space H. Does there exists a wandering subspace W for (T 1,..., T n )? This question has an affirmative answer for the restriction of multiplication operator by the coordinate function M z, to an invariant subspace of the Hardy space H 2 (D) (Beurlings theorem [Beu]) or the Bergman space A 2 (D) (Aleman, Richter and Sundberg [ARS]) or the Dirichlet space D(D) (Richter [Ric] ) over the unit disc D of the complex plane C. For n 2, existence of wandering subspaces for general invariant subspaces of the Hardy space H 2 (D n ) over the unit polydisc D n rather fails spectacularly (cf. [Rud]) Mathematics Subject Classification. 47A13, 47A15, 47A20, 47L99. Key words and phrases. Invariant subspace, Beurling s theorem, Bergman space, Dirichlet space, Hardy space, Doubly commutativity. 1

4 2 CHATTOPADHYAY, DAS, SARKAR, AND SARKAR Recall that the Hardy space over the unit polydisc D n = {z = (z 1,..., z n ) C n : z i < 1, i = 1,..., n} is denoted by H 2 (D n ) and defined by H 2 (D n ) = {f O(D n ) : sup f(rz) 2 dθ < }, 0 r<1 T n where dθ is the normalized Lebesgue measure on the torus T n, the distinguished boundary of D n, rz := (rz 1,..., rz n ) and O(D n ) denotes the set of all holomorphic functions on D n (cf. [Rud]). In [Beu], A. Beurling characterize all closed M z -invariant subspaces of H 2 (D) in the following sense: Let S = {0} be a closed subspace of H 2 (D). Then S is M z -invariant if and only if S = θh 2 (D) for some inner function θ (that is, θ H (D) and θ = 1 a.e. on the unit circle T). In particular, Beurlings theorem yields the wandering subspace property for M z -invariant subspaces of H 2 (D) as follows: if S = θh 2 (D) is an M z -invariant subspace of H 2 (D) then (1) S = m 0 z m W, where W is the wandering subspace for M z S given by W = S zs = θh 2 (D) zθh 2 (D) = θc. In [Rud], W. Rudin showed that there are invariant subspaces M of H 2 (D 2 ) which does not contain any bounded analytic function. In particular, the Beurling like characterization of (M z1,..., M zn )-invariant subspaces of H 2 (D n ), in terms of inner functions on D n is not possible. On the other hand, the wandering subspace (and the Beurlings theorem) for the shift invariant subspaces of the Hardy space H 2 (D) follows directly from the classical Wold decomposition [Wol] theorem for isometries. Indeed, for a closed M z -invariant subspace S( {0}) of H 2 (D), (M z S ) m S = Mz m S Mz m H 2 (D) = {0}. m=0 m=0 Consequently, by Wold decomposition theorem for the isometry M z S on S we have S = z m W ( z S ) m 0 m=0(m m S) = z m W, m 0 where W = S zs and hence, (1) follows. Therefore, the notion of wandering subspaces is stronger (as well as of independent interest) than the Beurlings characterization of M z - invariant subspaces of H 2 (D). To proceed, we now recall the following definition: a commuting n-tuple (n 2) of bounded linear operators (T 1,..., T n ) on H is said to be doubly commuting if T i Tj = Tj T i for all 1 i < j n. Natural examples of doubly commuting tuple of operators are the multiplication operators by the coordinate functions acting on the Hardy space or the Bergman space or the Dirichlet space over the unit polydisc D n (n 2). m=0

5 WANDERING SUBSPACES OF THE BERGMAN SPACE AND THE DIRICHLET SPACE OVER D n 3 Let H O(D n ) be a reproducing kernel Hilbert space over D n (see [Aro]) and multiplication operators {M z1,..., M zn } by coordinate functions are bounded. Then a closed (M z1,..., M zn )-invariant subspace S of H is said to be doubly commuting if the n-tuple (M z1 S,..., M zn S ) is doubly commuting, that is, R i Rj = RjR i for all 1 i < j n, where R i = M zi S. In [SSW], third author and Sasane and Wick proved that any doubly commuting invariant subspace of H 2 (D n ) (where n 2) has the wandering subspace property (see [Man] for n = 2). Also in [ReT], Redett and Tung obtained the same conclusion for doubly commuting invariant subspaces of the Bergman space A 2 (D 2 ) over the bidisc D 2. In this paper we prove that doubly commuting invariant subspaces of the Bergman space A 2 (D n ) and the Dirichlet space D(D n ) has the wandering subspace property. Our result on the Bergman space over polydisc is a generalization of the base case n = 2 in [ReT]. Our analysis is based on the Wold-type decomposition result of S. Shimorin for operators closed to isometries [Shi]. The paper is organized as follows: in Section 2, we obtain some general results concerning the wandering subspaces of tuples of doubly commuting operators. We obtain wandering subspaces for doubly commuting shift invariant subspaces of the Bergman and Dirichlet spaces over D n in Section Wandering subspace for tuple of doubly commuting operators In this section we prove the multivariate version of the S. Shimorin s result for tuple of doubly commuting operators on a general Hilbert space. We show that for a tuple of doubly commuting operators T = (T 1,..., T n ) on a Hilbert space H, if for any reducing subspace S i of T i the subspace S i T i S i is a wandering subspace for T i Si, i = 1,..., n, then n (H T i H) is a wandering subspace for T. We fix for the rest of the paper a natural number n 2 and set Λ n := {1,..., n}. For a closed subset K of a Hilbert space H, an n-tuple of commuting operators T = (T 1,..., T n ) on H and a non-empty subset α = {α 1,..., α k } Λ n, we write [K] Tα to denote the smallest closed joint T α := (T α1,..., T αk )-invariant subspace of H containing K. In other words (2) [K] Tα = l 1,l 2,...,l k =0 T l 1 α 1 T l 2 α 2 T l k α k (K). If α is a singleton set {i} then we simply write [K] Ti. For a non-empty subset α = {α 1,..., α k } Λ n, we also denote by W α subspace of H, the following k (3) W α = (H T αi H).

6 4 CHATTOPADHYAY, DAS, SARKAR, AND SARKAR Again if α is a singleton set {i} then we simply write W i. Thus with the above notation W α = W αi. α i α A bounded linear operator T on a Hilbert space H is analytic if concave if it satisfies the following inequality T 2 x 2 + x 2 2 T x 2 (x H). n=0 T n H = {0} and it is Multiplication by coordinate functions on the Dirichlet space over the unit polydisc are concave operators as we show in the next section. For a single bounded operator T on a Hilbert space H, the following result ensure the existence of the wandering subspace under certain conditions (see [Ric], [Shi]). Theorem 2.1. (Richter, Shimorin). Let T be an analytic operator on a Hilbert space H satisfies one of the following properties: (i) T x + y 2 2( x 2 + T y 2 ) (x, y H), (ii) T is concave. Then H T H is a wandering subspace for T, that is, H = [H T H] T. The following proposition is essential in order to generalize the above result for certain tuples of commuting operators. Proposition 2.2. Let T = (T 1,..., T n ) be a doubly commuting tuple of operators on H. Then W α is T j -reducing subspace for all non-empty subset α Λ n and j / α, where W α is as in (3). Proof. Let α = {α 1,..., α k } be a non-empty subset of Λ n and j / α. First note that W l = Ker Tl for all 1 l n and therefore W α = k Ker Tα i. Let x W α, α i α and y H. By doubly commutativity of T we have, T j x, T αi y = T α i T j x, y = T j T α i x, y = 0. Therefore, T j W α T αi H and hence T j W α W αi for all α i α. Thus W α is an invariant subspace for T j. Also, by commutativity of T we have T j x, T αi y = T α i T j x, y = T j T α i x, y = 0, for all α i α and y H. This implies Tj W α T αi H and then Tj W α W αi for all α i α. This completes the proof. Now we prove the main theorem in the general Hilbert space operator setting. Below for a set α, we denote by #α the cardinality of α.

7 WANDERING SUBSPACES OF THE BERGMAN SPACE AND THE DIRICHLET SPACE OVER D n 5 Theorem 2.3. Let T = (T 1,..., T n ) be a doubly commuting tuple of operators on H such that for any reducing subspace S i of T i, the subspace S i T i S i is a wandering subspace for T i Si, i = 1,..., n. Then for each non-empty subset α = {α 1,..., α k } Λ n, the tuple T α = (T α1,..., T αk ) has the wandering subspace property. Moreover, the corresponding wandering subspace is given by k W α = (H T αi H). Proof. First note that we only need to show H = [W α ] Tα for any non-empty subset α of Λ n as the orthogonal property for wandering subspace is immediate. Now for a singleton set α the result follows form the assumption that W i is a wandering subspace for T i, i = 1,..., n. Now for #α 2, it suffices to show that [W α ] Tαi = W α\{αi } for any α i α. Because for α i, α j α, one can repeat the procedure to get [W α ] T{αi,α j } = [[W α] Tαi ] Tαj = W α\{αi,α j }, and continue this process until the set α \ {α i, α j } becomes a singleton set and finally apply the assumption for singleton set. To this end, let α Λ n, #α 2 and α i α. Consider the set F = W α\{αi } T αi (W α\{αi }). Now by Proposition 2.2, W α\{αi } is a reducing subspace for T αi and therefore F = {x W α\{αi } : x Ker T α i } = W α\{αi } W αi = W α. On the other hand, since W α\{αi } is a reducing subspace for T αi then by assumption F = W α is a wandering subspace for T αi. Thus This completes the proof. [W α ] Tαi = W α\{αi }. Combining Theorem 2.1 with the above theorem we have the following result, which is the case of our main interest. Corollary 2.4. Let T = (T 1,..., T n ) be a commuting tuple of analytic operators on H such that T is doubly commuting and satisfies one of the following properties: (a) T i is concave for each i = 1,..., n (b) T i x + y 2 2( x 2 + T i y 2 ) (x, y H, i = 1, 2,..., n). Then for any non-empty subset α = {α 1,..., α k } Λ n, W α is a wandering subspace for T α = (T α1,..., T αk ), where W α is as in (3). Proof. Note that if T satisfies condition (a) or (b) then for any 1 i n and reducing subspace S i of T i, T i Si also satisfies condition (a) or (b) respectively. Thus by Theorem 2.1, T satisfies the hypothesis of the above theorem and the proof follows.

8 6 CHATTOPADHYAY, DAS, SARKAR, AND SARKAR We end this section by proving kind of converse of the above result and part of which is a generalization of [ReT], Theorem 3. Theorem 2.5. Let T = (T 1,..., T n ) be an n-tuple of commuting operators on H with the property T i x + y 2 2( x 2 + T i y 2 ) (x, y H, i = 1, 2,..., n) or T i is concave for all i = 1,..., n. Then, (i) T is doubly commuting on H, and (ii) T i is analytic for all i = 1, 2,..., n if and only if (a) for any non-empty subset α = {α 1,..., α k } Λ n, W α is a wandering subspace for T α = (T α1,..., T αk ) and for k 2, [W α ] Tαi = W α\{αi } for all α i α, (b) T i commutes with T j T j for all 1 i < j n. Proof. The forward direction follows from Theorem 2.3. For the converse, suppose that (a) and (b) hold. To show (i), let 1 i n be fixed. By assumption (a), W i = [W {i,j} ] Tj for all 1 i j n. This shows that W i is T j invariant subspace for all 1 j i n. Let x H. Since [W i ] Ti = H then there exists a sequence x k converging to x such that x k = N k m=0 T m i x m,k, where each N k is a nonnegative integer and x m,k is a member of W i. Now for any 1 j i n we have, On the other hand, T j T i x k = T i T j x k = N k m=0 N k m=0 T j T i T m i x m,k = T i T j T m i x m,k = N k m=1 N k m=1 T j T i T m i x m,k = T i T j T m i x m,k. N k m=1 T i T j T m i x m,k, where the last equality follows from (b). So Ti T j x k = T j Ti x k and by taking limit Ti T j x = T j Ti x. Thus we have (i). Finally, by Theorem 3.6 of [Shi] we have that for all 1 i n. By part (a), [W i ] Ti H = [W i ] Ti m=1 T m i H, = H for all 1 i n. Thus m=1 Ti m H = {0} for all 1 i n and this completes the proof.

9 WANDERING SUBSPACES OF THE BERGMAN SPACE AND THE DIRICHLET SPACE OVER D n 7 3. Wandering subspaces for invariant subspaces of A 2 (D n ) and D(D n ) In this section we apply the general theorem proved in the previous section to obtain wandering subspaces for invariant subspaces of the Bergman space and the Dirichlet space over polydisc. The Bergman space over D n is denoted by A 2 (D n ) and defined by A 2 (D n ) = {f O(D n ) : f(z) 2 da(z) < }, D n where da is the product area measure on D n. Below for any invariant subspace S of A 2 (D n ) and non-empty set α = {α 1,..., α k } Λ n, we denote by Wα S the following set: k Wα S := (S z αi S). We denote by M = (M z1,..., M zn ) the n-tuple of co-ordinate multiplication operators on A 2 (D n ). Theorem 3.1. Suppose S is a closed joint M-invariant subspace of A 2 (D n ). If S is doubly commuting then for any non-empty subset α = {α 1,..., α k } of Λ n, Wα S is a wandering subspace for M α S = (M zα1 S,..., M zαk S ). The proof of the above theorem follows if we show the tuple of operators (M z1 S..., M zn S ) on S satisfies all the hypothesis of Corollary 2.4. First note that by analyticity of A 2 (D n ), all the co-ordinate multiplication operators M zi, i = 1,..., n on A 2 (D n ) are analytic. Then M zi S is also analytic for all i = 1,..., n. Thus the only hypothesis remains to verify is either condition (a) or (b) of Corollary 2.4. In the next lemma we show that the tuple (M z1,..., M zn ) satisfies condition (b) and therefore so does (M z1 S..., M zn S ). Lemma 3.2. For all 1 i n the operators satisfies for all f, g A 2 (D n ). M zi : A 2 (D n ) A 2 (D n ), f z i f, ) M zi f + g 2 A 2 (D n ) ( f 2 2 A 2 (D n ) + M z i g 2 A 2 (D n ), Before we prove this lemma, we recall a well known fact regarding the norm of a function in A 2 (D n ) and prove an inequality. If f is in A 2 (D n ) with the following power series expansion corresponding to i-th variable: f(z 1,, z n ) = a k zi k,

10 8 CHATTOPADHYAY, DAS, SARKAR, AND SARKAR where a k A 2 (D n 1 ) for all k N, then f 2 A 2 (D n ) = a k 2 A 2 (D n 1 ). (k + 1) Next we prove the following inequality (can be found in [DuS], page 277, we include the proof for completeness) for any z, w C and k N \ {0}, ( ) z + w 2 z 2 (4) k k + w 2. k + 2 To this end, note that 2k(k + 2) Re(z w) (k + 2) 2 z 2 + k 2 w 2, which follows from the inequality (k + 2)z kw 2 0. Now z + w 2 k + 1 Now we prove the lemma. = z 2 + w Re(z w) k + 1 z 2 + w 2 + z 2 (k + 2)/k + w 2 k/(k + 2) k + 1 ( z 2 = 2 k + w 2 k + 2 ). k + 1 Proof. Let 1 i n be fixed. Let f(z 1,, z n ) = a k zi k and g(z 1,, z n ) = b k zi k be the power series expansions of two functions f, g A 2 (D n ) with respect to z i -th variable, where a k, b k A 2 (D n 1 ) for all k N. Then Now (M zi f + g) = a k z k+1 i + b k zi k = b 0 + (a k 1 + b k )zi k. M zi f + g 2 A 2 (D n ) = b 0 2 A 2 (D n 1 ) + a k 1 + b k 2 A 2 (D n 1 ) (k + 1) k=1 ( b 0 2 A 2 (D n 1 ) + 2 ak 1 2 A 2 (D n 1 ) + b ) k 2 A 2 (D n 1 ) ( by (4)) k k + 2 k=1 ( ) a k 2 A = 2 2 (D n 1 ) b k 1 2 A + 2 (D n 1 ) k + 1 k + 1 k=1 ( ) = 2 f 2 A 2 (D n ) + M z i g 2 A 2 (D n ). k=1 Thus the result.

11 WANDERING SUBSPACES OF THE BERGMAN SPACE AND THE DIRICHLET SPACE OVER D n 9 Now we turn our discussion to the Dirichlet space over polydisc. The Dirichlet space over D is denoted by D(D) and defined by D(D) := {f O(D) : f A 2 (D)}. For any f = a k z k D(D), f 2 D = (k + 1) a k 2. The Dirichlet space over D n is denoted by D(D n ) and defined by D(D n ) := D(D) D(D). }{{} n-times Another way of expressing D(D n ) is the following D(D n ) := {f O ( D; D(D n 1 ) ) : f = a k z k, (k + 1) a k 2 D(D n 1 ) < }. In the above, a k D(D n 1 ) for all k N. The co-ordinate multiplication operators on D(D n ) are also denoted by M zi, i = 1,..., n. Since D(D n ) contains holomorphic functions on D n then M zi is analytic for all i = 1,..., n. Let 1 i n be fixed. Now for f D(D n ), let f = a k zi k be the Taylor expansion of f corresponding to z i -th variable, where a k D(D n 1 ) for all k N. Then f 2 = (k + 1) a k 2 D(D n 1 ) and M 2 z i f 2 + f 2 = = = 2 (k + 1) a k 2 2 D(D n 1 ) + (k + 1) a k 2 D(D n 1 ) k=2 (k 1) a k 2 D(D n 1 ) + (k + 1) a k 2 D(D n 1 ) k a k 2 D(D n 1 ) k=1 = 2 M zi f 2. Therefore M zi is concave for all i = 1,..., n. Thus again by Corollary 2.4, we have proved the following result of wandering subspaces for invariant subspaces of Dirichlet space over polydisc. Theorem 3.3. Suppose S is a closed joint (M z1,..., M zn )-invariant subspace of D(D n ). If S is doubly commuting then for any non-empty subset α = {α 1,..., α k } of Λ n, Wα S is a wandering subspace for M α S = (M zα1 S,..., M zαk S ), where W S α = k (S z αi S).

12 10 CHATTOPADHYAY, DAS, SARKAR, AND SARKAR We conclude the paper with the remark that since the hypothesis of Corollary 2.4 and Theorem 2.5 are same then the same conclusion as in Theorem 2.5 holds for invariant subspaces of Bergman space or Dirichlet space over polydisc. Acknowledgment: First two authors are grateful to Indian Statistical Institute, Bangalore Centre for warm hospitality. The fourth author was supported by UGC Centre for Advanced Study. References [Aro] N. Aronszajn, Theory of reproducing kernels, Trans. of American Mathematical Soc. 68 (1950), [ARS] A. Aleman, S. Richter and C. Sundberg, Beurling s theorem for the Bergman space, Acta. Math. 177 (1996), [Beu] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta. Math. 81 (1949), [DuS] P. Duren and A. Schuster, Bergman Spaces, Math. Surveys and Monographs, 100, Amer. Math. Soc. Providence. [Hal] P. R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961) [Man] V. Mandrekar, The validity of Beurling theorems in polydiscs, Proc. Amer. Math. Soc. 103 (1988), [ReT] D. Redett and J. Tung, Invariant subspaces in Bergman space over the bidisc Proc. Amer Math Soc, 138 (2010), [Ric] S. Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386 (1988), [Rud] W. Rudin, Function theory in polydiscs, Benjamin, New York, [Shi] S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001), [SSW] J. Sarkar, A. Sasane and B. Wick, Doubly commuting submodules of the Hardy module over polydiscs, preprint, arxiv: [Wol] H. Wold, A study in the analysis of stationary time series, Almquist and Wiksell, Uppsala, (A. Chattopadhyay) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, , India address: arup@isibang.ac.in, 2003arupchattopadhyay@gmail.com (B. K. Das) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, , India address: dasb@isibang.ac.in, bata436@gmail.com (J. Sarkar) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, , India address: jay@isibang.ac.in, jaydeb@gmail.com (S. Sarkar) Department of Mathematics, Indian Institute of Science, Bangalore, , India address: santanu@math.iisc.ernet.in