On quasi-similarity of multiplication operator on the weighted Bergman space in the unit ball
aa r X i v : . [ m a t h . F A ] F e b ON QUASI-SIMILARITY OF MULTIPLICATION OPERATOR ONTHE WEIGHTED BERGMAN SPACE IN THE UNIT BALL
CUI CHEN ∗ , YA WANG AND YU-XIA LIANG Abstract.
For α > −
1, let A α ( B N ) be the weighted Bergman space on theunit ball B N in C N . In this paper, we prove that the multiplication op-erator M z n is quasi-similar to ⊕ Q Ni =1 n i M z on A α ( B N ) for the multi-index n = ( n , n , · · · , n N ). Introduction
Similarity of operators is a weaker concept than unitary equivalence. It is wellknown that invariant subspace of an operator can be identified in terms of invariantsubspaces of a similar operator. Over the years, there are some results characteriz-ing the lattice of closed invariant subspaces by constructing similar operators, werefer the interested readers to the recent papers such as [2, 3, 5].In previous years, the similarity between M z n and ⊕ n M z acting on a Hilbertspace is an active topic which has be concerned in lots of papers. In 2007, Jiangand Li (see [8]) first obtained that analytic Toeplitz operator M B ( z ) is similar to ⊕ n M z on the Bergman space if and only if B ( z ) is an n -Blaschke product. Next,Li (see [10]) in 2009 proved that multiplication operator M z n is similar to ⊕ n M z on the weighted Bergman space. And then, Jiang and Zheng in [9] extended themain result in [8] to the weighted Bergman space. In 2011, Douglas and Kim in [6]investigated the reducing subspaces for an analytic multiplication operator M z n onthe Bergman space A α ( A r ) of the annulus A r . The similarity of M z n and ⊕ n M z also holds on the weighted Hardy space and Sobolev disk algebra, they are shownin [1] and [12]. For further results related, see [7,13]. Moreover, in 2017, the unitaryequivalence of analytic multipliers on Sobolev disk algebra was discussed in [4]. Butsuch a characterization is not always hold for all Hilbert spaces. In Recent paper,Li, Lan and Liu (see [11]) proved that multiplication operator M z n is not similarto ⊕ n M z on the Fock space, they are quasi-similar, actually.However, all these results are considered in the one-dimensional space, such acharacterization is much more complicated for the high-dimension case. Based onthe works above, in this paper we are interested in the corresponding result on theweighted Bergman space in the unit ball.Let B N be the unit ball in C N , where C N is the N -dimensional complex vectorspace. Denoted by N and N the set of all positive and nonnegative integers,respectively. H ( B N ) is the class of all holomorphic functions on B N . Mathematics Subject Classification.
Primary: 47B38; Secondary: 32H02, 30H05, 30H20,47B33.
Key words and phrases. multiplication operator, weighted Bergman space, quasi-similarity,unit ball. ∗ Corresponding author.
Given three multi-indexes in N N with n = ( n , n , · · · , n N ) , k = ( k , k , · · · , k N )and j = ( j , j , · · · , j N ), we use the notations | n | = n + n + · · · + n N , n ! = n ! n ! · · · n N ! . and nk + j = ( n k + j , n k + j , · · · , n N k N + j N ) . Moreover, we say j < (or ≤ ) n if j i < (or ≤ ) n i for each i = 1 , , · · · , N .For z ∈ C N , z n = z n z n · · · z n N N . Let α > −
1, the weighted Bergman space A α ( B N ) consists of holomorphic func-tions f in L ( B N , dv α ), that is, A α ( B N ) = L ( B N , dv α ) ∩ H ( B N ). It is well knownthat A α ( B N ) is a Hilbert space with the inner product defined by h f, g i = Z B N f gdv α , for f, g ∈ A α ( B N ) . The corresponding norm of f is given by k f k = R B N | f | dv α . As we know, the setof polynomials is dense in the Bergman space A α ( B N ), and moreover, h z k , z m i = Z B N z k z m dv α = ( k !Γ( N + α +1)Γ( N + | k | + α +1) , m = k, , otherwise. Thus { e k ( z ) , q Γ( N + | k | + α +1) k !Γ( N + α +1) z k : k ∈ N N } ia an orthonormal basis of A α ( B N ).Throughout the rest of this paper, we will always assume that N ≥ N = 1. Moreover, we would like topoint out that n stands for the multi-index ( n , n , · · · , n N ) ∈ N N with n i ≥ i ∈ { , , · · · , N } , as it is trivial for otherwise.2. Quasi-similarity
Let us recall that for two Hilbert spaces H and K , an operator X ∈ L ( H , K )is said to be quasi-invertible if it has zero kernel and dense range. Let S ∈ L ( H )and T ∈ L ( K ), S is quasi-similar to T if there exist two quasi-invertible operators X ∈ L ( H , K ) and Y ∈ L ( K , H ) respectively such that XS = T X and SY = Y T .The following theorem is our main result.
Theorem 1. M z n is quasi-similar to ⊕ Q Ni =1 n i M z acting on the weighted Bergmanspace A α ( B N ) with α > − . Before going to the main theorem, we need the following lemma.
Lemma 1.
For a multi-index j = ( j , j , · · · , j N ) , j i = 0 , , · · · , n i − for each i =1 , , · · · , N , let A j = span { e nk + j : k = ( k , k , · · · , k N ) ∈ N N } . Then(i) { e nk + j : k = ( k , k , · · · , k N ) ∈ N N } form an orthonormal basis of A j .(ii) A α ( B N ) = ⊕ j A j , where the direct sum is over all multi-indexes j = ( j , j , · · · , j N ) and j i = 0 , , · · · , n i − for each i = 1 , , · · · , N .(iii) A j is a reducing subspace for M z n . Proof. (i) Note that for all k, m ∈ N N , h e nk + j , e nm + j i = Z B s Γ( N + | nk + j | + α + 1)( nk + j )!Γ( N + α + 1) z nk + j s Γ( N + | nm + j | + α + 1)( nm + j )!Γ( N + α + 1) z nm + j dv α ( z ) . UASI-SIMILARITY OF MULTIPLICATION OPERATOR 3 If m = k , then h e nk + j , e nm + j i = 0. If m = k , we have h e nk + j , e nm + j i = Γ( N + | nk + j | + α + 1)( nk + j )!Γ( N + α + 1) Z B N | z nk + j | dv α ( z )= Γ( N + | nk + j | + α + 1)( nk + j )!Γ( N + α + 1) ( nk + j )!Γ( N + α + 1)Γ( N + | nk + j | + α + 1) = 1 . Thus (i) holds.(ii) It is clear that A j ⊥ A t for all 0 ≤ j = t ≤ n −
1. Next, for f ∈ A α ( B N ), itis easy to see that f has the form f = ∞ X k =0 n − X j =0 a jk e nk + j . Suppose f = 0, we conclude that a jk = 0 for all k ∈ N N and j = ( j , j , · · · , j N ) , j i =0 , , · · · , n i −
1, since h ∞ X k =0 n − X j =0 a jk e nk + j , e l i = 0 for each l ∈ N N . That is 0 = Π Ni =1 n i z }| { ⊕ ⊕ · · · ⊕
0, which yields that A α ( B N ) = ⊕ j A j .(iii) It is easy to see that both A j and A ⊥ j are invariant subspaces for M z n . (cid:3) By the previous lemma, it is clear that M z n = ⊕ j M z n | A j . Then we can get theproof of Theorem 1. Proof of Theorem 1.
Note that M z e k = z s Γ( N + | k | + α + 1) k !Γ( N + α + 1) z k = s ( k + 1)Γ( N + | k | + α + 1)Γ( N + | k + 1 | + α + 1) e k +1 . Set M j = M z n | A j , then M j e nk + j = z n s Γ( N + | nk + j | + α + 1)( nk + j )!Γ( N + α + 1) z nk + j = s ( n ( k + 1) + j )!Γ( N + | nk + j | + α + 1)( nk + j )!Γ( N + | n ( k + 1) + j | + α + 1) e n ( k +1)+ j Define X j : A α ( B N ) → A j such that X j e k = c kj e nk + j , where c kj are given by c kj = s ( nk + j )!Γ( N + | k | + α + 1)Γ( N + | j | + α + 1) k ! j !Γ( N + | nk + j | + α + 1)Γ( N + α + 1) . Thus we conclude that X j M z e k = M j X j e k . In fact, X j M z e k = X j s ( k + 1)Γ( N + | k | + α + 1)Γ( N + | k + 1 | + α + 1) e k +1 = s ( nk + n + j )!Γ( N + | k | + α + 1)Γ( N + | j | + α + 1) k ! j !Γ( N + | nk + n + j | + α + 1)Γ( N + α + 1) e nk + n + j , C. CHEN, Y. WANG AND Y.X. LIANG and M j X j e k = M j s ( nk + j )!Γ( N + | k | + α + 1)Γ( N + | j | + α + 1) k ! j !Γ( N + | nk + j | + α + 1)Γ( N + α + 1) e nk + j = s ( nk + n + j )!Γ( N + | k | + α + 1)Γ( N + | j | + α + 1) k ! j !Γ( N + | nk + n + j | + α + 1)Γ( N + α + 1) e nk + n + j . Now we will show that X j is not invertible in the following by giving the fact thatlim inf | k |→∞ c kj = 0. Indeed, c kj = Γ( N + | j | + α + 1) j !Γ( N + α + 1) | nk + j | !Γ( N + | k | + α + 1) | k | !Γ( N + | nk + j | + α + 1) ( nk + j )! | k | ! | nk + j | ! k ! , Γ( N + | j | + α + 1) j !Γ( N + α + 1) I I , (1)where I = | nk + j | !Γ( N + | k | + α + 1) | k | !Γ( N + | nk + j | + α + 1) and I = ( nk + j )! | k | ! | nk + j | ! k ! . For I , | nk + j | !Γ( N + | k | + α + 1) | k | !Γ( N + | nk + j | + α + 1)= | nk + j | ( | nk + j | − · · · ( | k | + 1)( N + | nk + j | + α )( N + | nk + j | + α − · · · ( N + | k | + α + 1)= | k | + 1 N + | nk + j | + α ( | nk + j | )( | nk + j | − · · · ( | k | + 2)( N + | nk + j | + α − · · · ( N + | k | + α + 1)= | k | + 1 N + | nk + j | + α N + α − | nk + j | )(1 + N + α − | nk + j |− ) · · · (1 + N + α − | k | +2 ) (2)Define a kj ( α ) = (1 + N + α − | nk + j | )(1 + N + α − | nk + j | − · · · (1 + N + α − | k | + 2 ) . Since α > − N ≥
2, it is clear N + α − >
0. Then it easily follows that a kj ( α ) ≤ (1 + N + α − | k | + 2 ) | nk + j |−| k |− ≤ (1 + N + α − | k | + 2 ) | n || k | + | j |−| k |− = (1 + N + α − | k | + 2 ) | k | +2 N + α − · N + α − | k | +2 · [ | n || k | + | j |−| k |− → exp { ( | n | − N + α − } , as | k | → ∞ . (3) UASI-SIMILARITY OF MULTIPLICATION OPERATOR 5
On the other hand, a kj ( α ) ≥ (1 + N + α − | nk + j | ) | nk + j |−| k |− ≥ (1 + N + α − | n || k | + | j | ) | k | + | j |−| k |− = (1 + N + α − | n || k | + | j | ) | j |− → , as | k | → ∞ . (4)From (2)-(4), associated with1 | n | = lim | k |→∞ | k | + 1 N + | n || k | + | j | + α ≤ lim | k |→∞ | k | + 1 N + | nk + j | + α ≤ lim | k |→∞ | k | + 1 N + n m | k | + α = 1 n m , where m ∈ { , , · · · , N } is the index such that n m = min { n , n , · · · , n N } , thenwe conclude that there exist two positive constants C and C such that C ≤ lim inf | k |→∞ I ≤ lim sup | k |→∞ I ≤ C . (5)For I , first note that( n k + j )! · · · ( n N k N + j N )!( n k + j + · · · + n N k N + j N )! ≤ ( n m k + j m )! · · · ( n m k N + j m )!( n m k + j m + · · · + n m k N + j m )! . (6)This is due to that( n k + j )! · · · ( n N k N + j N )!( n m k + j m )! · · · ( n m k N + j m )! · ( n m k + j m + · · · + n m k N + j m )!( n k + j + · · · + n N k N + j N )!= Π Ni =1 [( n i k i + j i )( n i k i + j i − · · · ( n m k i + j m + 1)] P Ni =1 ( n i k i + j i ) { P Ni =1 ( n i k i + j i ) − } · · · { P Ni =1 ( n m k i + j m ) + 1 } ≤ , which comes from the numbers of factors are both P Ni =1 ( n i k i + j i − n m k i − j m )in the numerator and denominator, and the molecular factors are not greater thanthe denominator factors one by one.Using Stirling’s approximation, we havelim inf | k |→∞ I ≤ lim k = k = ··· = k N →∞ I = lim k →∞ [( n m k + j m )!] N [ N ( n m k + j m )]! ( N k )!( k !) N = lim k →∞ [( n m k + j m ) n m k + j m √ n m k + j m e − ( n m k + j m ) ] N [ N ( n m k + j m )] N ( n m k + j m ) p N ( n m k + j m ) e − N ( n m k + j m ) · ( N k ) Nk √ N k e − Nk [ k k √ k e − k ] N = lim k →∞ n ( N − / m ( N N ) (1 − n m ) k − j m Thus if n m >
1, lim inf | k |→∞ I = 0. C. CHEN, Y. WANG AND Y.X. LIANG
For the remaining case n m = 1. Without loss of generality, we may assume that ♯ { i ∈ { , , · · · , N } : n i = 1 } = 1, for otherwise set k m ′ = 0 when n m ′ = 1. Choose s such that n s = min { n i : i ∈ { , , · · · , N }\{ m }} . From the same reason of (6) wecan get ( n k + j )! · · · ( n N k N + j N )!( n k + j + · · · + n N k N + j N )! ≤ ( n s k + j s )! · · · ( n s k m − + j s )!( k m + j m )!( n s k m +1 + j s )! · · · ( n s k N + j s )!( n s k + j s + · · · + n s k m − + j s + k m + j m + n s k m +1 + j s + · · · + n s k N + j s )! . By Stirling’s approximation, we conclude thatlim inf | k |→∞ I ≤ lim k = k = ··· = k N →∞ I = lim k →∞ [( n s k + j s )!] N − ( k + j m )![( N − n s k + j s ) + k + j m ]! ( N k )!( k !) N = lim k →∞ ( n s k + j s ) ( N − n s k + j s ) p ( n s k + j s ) N − e − ( N − n s k + j s ) [( N − n s k + j s ) + k + j m ] ( N − n s k + j s )+ k + j m · ( k + j m ) k + j m √ k + j m e − ( k + j m ) p ( N − n s k + j s ) + k + j m e − [( N − n s k + j s )+ k + j m ] · ( N k ) Nk √ N k e − Nk ( k k √ k e − k ) N = lim k →∞ N Nk ( N − k + j m n s k + j s ) ( N − n s k + j s ) (1 + ( N − n s k + j s ) k + j m ) k + j m · p ( n s k + j s ) N − √ k + j m √ N k p ( N − n s k + j s ) + k + j m p k N = lim k →∞ (cid:16) N ( N − n s ) n s (cid:17) ( N − k (cid:16) N N − n s (cid:17) k s N n N − s ( N − n s + 1 · N − n s ) j s (1 + ( N − n s ) j m = 0 , which we have used (cid:16) N − n s (cid:17) n s > N and 1 + ( N − n s > N in the lastequation, since N, n s ≥
2. Along with (1) and (5), we have lim inf | k |→∞ c k,j = 0.This implies that X j is not invertible.Next, define Y j : A j → A α ( B N ) by Y j e nk + j = b kj e k , where b kj = c kj . It iseasy to prove that Y j M j = M z Y j on A j . From the above discussion, we havelim sup | k |→∞ b kj = ∞ . Hence, Y j is also not invertible.For the rest of the proof we will show that both X j , Y j are quasi-invertibleoperators. To do this, take f ∈ ker X j with f = P ∞ k =0 d k e k , d k ∈ C N . From0 = h X j f, e nk + j i = h P ∞ k =0 d k c kj e nk + j , e nl + j i , we have d l = 0 for all l ∈ N N , Hence X j is injective. Besides, pick g ∈ ker X ∗ j with g = P ∞ k =0 m k e nk + j , m k ∈ C N . From0 = h e l , X ∗ j g i = h c lj e nl + j , P ∞ k =0 m k e nk + j i , we have m l = 0 for all l ∈ N N . Then(Ran X j ) ⊥ = ker X ∗ j = { } , so Ran X j = A j , that is X j has dense range. Similarly,we can check that Y j is injective and has dense range. It turns out that both X j and UASI-SIMILARITY OF MULTIPLICATION OPERATOR 7 Y j are quasi-invertible. To sum up, M j is quasi-similar to M z . Since M z n = ⊕ j M j ,we conclude that M z n is quasi-similar to ⊕ Q Ni =1 n i M z , and we are done. (cid:3) References [1] M.F. Ahmadi and K. Hedayatian, On similarity of powers of shift operators, Turk. J. Math.,36 (2012) 596-600.[2] M.R. Alfonso, P.E. Manuel, Invariant subspaces of parabolic self-maps in the Dirichlet space,J. Funct. Anal., 266 (2014) 4115-4120.[3] M.R. Alfonso, P.E. Manuel and A.S. Stanislav, Invariant subspaces of parabolic self-maps inthe Hardy space, Math. Res. Lett., 17(1) (2010) 99-107.[4] Y. Chen, C. Qin and Q. Wu, Reducibility and unitary equivalence of analytic multipliers onSobolev disk algebra, J. Math. Anal. Appl., 455(2) (2017) 1249-1256.[5] ˇ Z . ˇ C uˇ c kovi´ c and B. Paudyal, Invariant subspaces of the shift plus complex Volterra operator,J. Math. Anal. Appl., 426 (2015) 1174-1181.[6] R.G. Douglas and Y.S. Kim, Reducing subspaces on the annulus, Integr. Equat. Oper. Th.,70 (2011) 1-15.[7] K. Ji and R. Shi, Similarity of multiplication operators on the Sobolev disk algebra, Acta.Math. Sin. Eng. Sers., 29(4) (2013) 789-800.[8] C.L. Jiang and Y.C. Li, The commutant and similarity invariant of analytic Toeplitz operatorson Bergman space, Sci. China Ser. A, 50(5) (2007) 651-664.[9] C.L. Jiang and D.C. Zheng, Similarity of analytic Toeplitz operators on the Bergman spaces,J. Funct. Anal., 258 (2010) 2961-2982.[10] Y.C. Li, On similarity of multiplication operator on weighted Bergman space, Integr. Equat.Oper. Th., 63 (2009) 95-102.[11] Y.C. Li, W.H. Lan and J.L. Liu, On quasi-similarity and renducing subspaces of multiplicationoperator on the Fock space, J. Math. Anal. Appl., 409 (2014) 899-905.[12] Y.C. Li, Q.J. Liu, and W.H. Lan, On similarity and renducing subspaces of multiplicationoperator on Sobolev disk algebra, J. Math. Anal. Appl., 419 (2014) 1161-1167.[13] R.F. Zhao, A similarity invariant and the commutant of some multiplication operators on theSobolev disk algebra, Int. J. Math. Math. Sci., Artical ID 378217, (2012) 17p. Cui Chen, Department of Mathematics, Tianjin University of Finance and Economics,Tianjin 300222, P.R. China
Email address : chencui [email protected] Ya Wang, Department of Mathematics, Tianjin University of Finance and Economics,Tianjin 300222, P.R. China
Email address : [email protected] Yu-Xia Liang, School of Mathematical Sciences, Tianjin Normal University, Tianjin300378, P.R. China.
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