Radial operators on polyanalytic weighted Bergman spaces
Roberto Moisés Barrera-Castelán, Egor A. Maximenko, Gerardo Ramos-Vazquez
aa r X i v : . [ m a t h . F A ] N ov Radial operators on polyanalyticweighted Bergman spaces
Roberto Mois´es Barrera-Castel´an, Egor A. Maximenko,Gerardo Ramos-VazquezNovember 4, 2020
Abstract
Let µ α be the Lebesgue plane measure on the unit disk with the radial weight α +1 π (1 − | z | ) α . Denote by A n the space of the n -analytic functions on the unit disk,square-integrable with respect to µ α . Extending the results of Ramazanov (1999,2002), we explain that disk polynomials (studied by Koornwinder in 1975 and W¨unschein 2005) form an orthonormal basis of A n . Using this basis, we provide the Fourierdecomposition of A n into the orthogonal sum of the subspaces associated with differentfrequencies. This leads to the decomposition of the von Neumann algebra of radialoperators, acting in A n , into the direct sum of some matrix algebras. In other words,all radial operators are represented as matrix sequences. In particular, we represent inthis form the Toeplitz operators with bounded radial symbols, acting in A n . Moreover,using ideas by Engliˇs (1996), we show that the set of all Toeplitz operators withbounded generating symbols is not weakly dense in B ( A n ).AMS Subject Classification (2010): Primary 22D25; Secondary 30H20, 47B35, 33C45.Keywords: radial operator, polyanalytic function, weighted Bergman space, meanvalue property, reproducing kernel, von Neumann algebra, Jacobi polynomial, diskpolynomial. Background
Polyanalytic functions naturally arise in some physical models (plane elasticity theory,Landau levels) and in some methods of signal processing, see [1–3,15,17,18]. Koshelev [23]computed the reproducing kernel of the n -analytic Bergman space A n ( D ) on the unit disk.Balk in the book [4] explained fundamental properties of polyanalytic functions. Dzhu-raev [9] related polyanalytic projections with singular integral operators. Vasilevski [40,41]studied polyanalytic Bergman spaces on the upper halfplane and polyanalytic Fock spacesusing the Fourier transform. Ramazanov [29, 30] constructed an orthonormal basis in A n ( D ) and studied various properties of A n ( D ). In fact, the elements of this basis arewell-known disk polynomials studied by Koornwinder [21], W¨unsche [43], and other au-thors. Pessoa [27] related Koshelev’s formula with Jacobi polynomials and gave a very1lear proof of this formula. He also obtained similar results for some other one-dimensionaldomains. Hachadi and Youssfi [14] developed a general scheme for computing the repro-ducing kernels of the spaces of polyanalytic functions on radial plane domains (disks orthe whole plane) with radial measures.There are general investigations about bounded linear operators in reproducing ker-nel Hilbert spaces (RKHS), especially about Toeplitz operators in Bergman or Fockspaces [5, 44, 45], but the complete description of the spectral properties is found onlyfor some special classes of operators, in particular, for Toeplitz operators with generat-ing symbols invariant under some group actions, see Vasilevski [42], Grudsky, Quiroga-Barranco, and Vasilevski [11], Dawson, ´Olafsson, and Quiroga-Barranco [8]. The simplestclass of this type consists of Toeplitz operators with bounded radial generating symbols.Various properties of these operators (boundedness, compactness, and eigenvalues) havebeen studied by many authors, see [13, 22, 28, 46]. The C*-algebra generated by such op-erators, acting in the Bergman space, was explicitly described in [6, 12, 16, 37]. Loaiza andLozano [24] obtained similar results for radial Toeplitz operators in harmonic Bergmanspaces. Maximenko and Teller´ıa-Romero [26] studied radial operators in the polyanalyticFock space.Hutn´ık, Hutn´ıkov´a, Ram´ırez-Mora, Ram´ırez-Ortega, S´anchez-Nungaray, Loaiza, andother authors [19, 20, 25, 31, 32, 35] studied vertical and angular Toeplitz operators in poly-analytic and true-polyanalytic Bergman spaces. In particular, vertical Toeplitz operatorsin the n -analytic Bergman space over the upper half-plane are represented in [31] as n × n matrices whose entries are continuous functions on (0 , + ∞ ), with some additional prop-erties at 0 and + ∞ .Rozenblum and Vasilevski [33] investigated Toeplitz operators with distributional sym-bols and showed that Toeplitz operators in true-polyanalytic spaces Bergman or Fockspaces are equivalent to some Toeplitz operators with distributional symbols in the ana-lytic Bergman or Fock spaces. Objects of study
Denote by µ the Lebesgue plane measure and its restriction to the unit disk D , and by µ α the weighted Lebesgue plane measured µ α ( z ) := α + 1 π (1 − | z | ) α d µ ( z ) . This measure is normalized: µ α ( D ) = 1. We use notation h· , ·i and k · k for the innerproduct and the norm in L ( D , µ α ).Let A n ( D , µ α ) be the space of n -analytic functions square integrable with respect to µ α . We denote by A n ) ( D , µ α ) the orthogonal complement of A n − ( D , µ α ) in A n ( D , µ α ).For every τ in the unit circle T := { z ∈ C : | z | = 1 } , let ρ ( α ) n ( τ ) be the rotation operatoracting in A n ( D , µ α ) by the rule ( ρ ( α ) n ( τ ) f )( z ) := f ( τ − z ) . ρ ( α ) n is a unitary representation of the group T in the space A n ( D , µ α ). Wedenote by R ( α ) n its commutant, i.e., the von Neumann algebra that consists of all boundedlinear operators acting in A n ( D , µ α ) that commute with ρ ( α ) n ( τ ) for every τ in T . In otherwords, the elements of R ( α ) n are the operators intertwining the representation ρ ( α ) n . Theelements of R ( α ) n are called radial operators in A n ( D , µ α ).In a similar manner, we denote by ρ ( α )( n ) ( τ ) the rotation operators acting in A n ) ( D , µ α )and by R ( α )( n ) the von Neumann algebra of radial operators in A n ) ( D , µ α ). We also considerthe rotation operators ρ ( α ) ( τ ) in L ( D , µ α ) and the corresponding algebra R ( α ) of radialoperators. Structure and results of this paper • In Section 2, we list some necessary facts about Jacobi polynomials. They play acrucial role in Sections 3 and 4. • In Section 3, we recall various equivalent formulas for the disk polynomials that canbe obtained by orthogonalizing the monomials in z and z . Using this orthonormalbasis ( b ( α ) p,q ) p,q ∈ N we descompose L ( D , µ α ) into the orthogonal sum of subspaces W ( α ) ξ corresponding to different frequences ξ , ξ ∈ Z . • In Section 4 we give an elementary proof of the weighted mean value property ofpolyanalytic functions and show the boundedness of the evaluation functionals forthe spaces of polyanalytic functions over general domains in C . In the unweightedcase, this mean value property was proven by Koshelev [23] and Pessoa [27]. Inthe weighted case, it was found by Hachadi and Youssfi [14] and used by them tocompute the reproducing kernel of A n ( D , µ α ). • In Section 5, extending results by Ramazanov [29,30] to the weighted case, we verifythat the family ( b ( α ) p,q ) p ≥ , ≤ q 2, whereas R ( α )( n ) is commutative for every n in N . • In Section 8, we find explicit representations of the radial Toeplitz operators actingin the spaces A n ( D , µ α ) and A n ) ( D , µ α ). The results of Sections 7 and 8 are sim-ilar to [26]. The main difference is that the orthonormal bases are given by otherformulas. 3ost of the facts in the Sections 2–5 are not new. We recall them in a self-contained form,for reader’s convenience.We hope that this paper can serve as a basis for further investigations about polyana-lytic or polyharmonic Bergman spaces and operators acting on these spaces. For example,an interesting task is to describe the C*-algebra generated by radial Toeplitz operatorsacting in A n ( D , µ α ). In this section, we recall some necessary facts about Jacobi polynomials. Most of themare explained in [38, Chapter 4]. For every α and β in R , the Jacobi polynomials can bedefined by Rodrigues formula: P ( α,β ) n ( x ) := ( − n n n ! (1 − x ) − α (1 + x ) − β d n d x n (cid:16) (1 − x ) n + α (1 + x ) n + β (cid:17) . (1)This definition and the general Leibniz rule imply an expansion into powers of x − x + 1: P ( α,β ) n ( x ) = n X k =0 (cid:18) n + αn − k (cid:19) (cid:18) n + βk (cid:19) (cid:18) x − (cid:19) k (cid:18) x + 12 (cid:19) n − k . (2)Formula (2) yields a symmetry relation, the values at the points 1 and − 1, and a formulafor the derivative: P ( α,β ) n ( − x ) = ( − n P ( β,α ) n ( x ) , (3) P ( α,β ) n (1) = (cid:18) n + αn (cid:19) , P ( α,β ) n ( − 1) = ( − n (cid:18) n + βn (cid:19) , (4) (cid:0) P ( α,β ) n (cid:1) ′ ( x ) = α + β + n + 12 P ( α +1 ,β +1) n − ( x ) . (5)With the above properties, it is easy to compute the derivatives of P ( α,β ) n at the point 1.Now Taylor’s formula yields another explicit expansion for P ( α,β ) n : P ( α,β ) n ( x ) = n X k =0 (cid:18) α + β + n + kk (cid:19)(cid:18) α + nn − k (cid:19) (cid:18) x − (cid:19) k . (6)If α > − β > − 1, then (6) can be rewritten as P ( α,β ) n ( x ) = Γ( α + n + 1) n ! Γ( α + β + n + 1) n X k =0 (cid:18) nk (cid:19) Γ( α + β + n + k + 1)Γ( α + k + 1) (cid:18) x − (cid:19) k . (7)For α > − β > − 1, we equip ( − , 1) with the weight (1 − x ) α (1 + x ) β , then denoteby h· , ·i ( − , ,α,β the corresponding inner product: h f, g i ( − , ,α,β := Z − f ( x ) g ( x ) (1 − x ) α (1 + x ) β d x. L (( − , , (1 − x ) α (1 + x ) β ) is a Hilbert space, and the set P of the univariatepolynomials is its dense subset. Using (1) and integrating by parts, for every f in P weget h f, P ( α,β ) n i ( − , ,α,β = 12 n h f ′ , P ( α +1 ,β +1) n − i ( − , ,α +1 ,β +1 . (8)Applying (8) and induction, it is easy to prove that the sequence ( P ( α,β ) n ) ∞ n =0 is an orthog-onal basis of L (( − , , (1 − x ) α (1 + x ) β ), that is, for every polynomial h of degree lessthan n , Z − h ( x ) P ( α,β ) n ( x )(1 − x ) α (1 + x ) β d x = 0 . (9)Furthermore, h P ( α,β ) m , P ( α,β ) n i ( − , ,α,β = 2 α + β +1 Γ( n + α + 1)Γ( n + β + 1)(2 n + α + β + 1)Γ( n + α + β + 1) n ! δ m,n . (10)Formulas (1) and (9), and induction allow us to compute the following integral for β > Z − P ( α,β +1) n ( x ) (1 − x ) α (1 + x ) β d x = 2 α + β +1 ( − n B( α + n + 1 , β + 1) , (11)where B is the well-known Beta function. Jacobi polynomials for the unit interval The function t t − , 1) onto ( − , Q ( α,β ) n the“shifted Jacobi polynomial” obtained from P ( α,β ) n by composing it with this change ofvariables: Q ( α,β ) n ( t ) := P ( α,β ) n (2 t − . The properties of Q ( α,β ) n follow easily from the properties of P ( α,β ) n . In particular, here areanalogs of (1), (6), and (7): Q ( α,β ) n ( t ) = ( − n n ! (1 − t ) − α t − β d n d t n (cid:16) (1 − t ) n + α t n + β (cid:17) , (12) Q ( α,β ) n ( t ) = n X k =0 (cid:18) α + β + n + kk (cid:19)(cid:18) β + nn − k (cid:19) ( − n − k t k . (13) Q ( α,β ) n ( t ) = Γ( n + β + 1) n ! Γ( n + α + β + 1) n X k =0 (cid:18) nk (cid:19) Γ( α + β + n + k + 1)Γ( β + k + 1) ( − n − k t k . (14)The sequence ( Q ( α,β ) n ) ∞ n =0 is orthogonal on (0 , 1) with respect to the weight (1 − t ) α t β , and Z Q ( α,β ) m ( t ) Q ( α,β ) n ( t )(1 − t ) α t β d t = δ m,n Γ( n + α + 1)Γ( n + β + 1)(2 n + α + β + 1)Γ( n + α + β + 1) n ! . (15)5lso, here are analogs of (9) and (11): Z h ( t ) Q ( α,β ) n ( t )(1 − t ) α t β d t = 0 , (16) Z Q ( α,β +1) n ( t )(1 − t ) α t β d t = ( − n B( α + n + 1 , β + 1) . (17)Substituting in (12) t by tu and applying the chain rule, we get ∂ n ∂t n (cid:16) (1 − tu ) n + α t n + β (cid:17) = n ! (1 − tu ) α t β Q ( α,β ) n ( tu ) . (18)Inspired by (15) we define the function J ( α,β ) n on (0 , 1) as J ( α,β ) n ( t ) := c ( α,β ) n (1 − t ) α/ t β/ Q ( α,β ) n ( t ) , (19)where c ( α,β ) n := s (2 n + α + β + 1) Γ( n + α + β + 1) n !Γ( n + α + 1)Γ( n + β + 1) . (20)Then Z J ( α,β ) m ( t ) J ( α,β ) n ( t ) d t = δ m,n . (21) Reproducing property for the polynomials on the unit interval Given n in N and α, β > − 1, we denote by R ( α,β ) n the polynomial R ( α,β ) n ( t ) := ( − n B( α + 1 , β + 1)B( α + n + 1 , β + 1) Q ( α,β +1) n ( t ) . (22) Proposition 2.1. Let n ∈ N and α, β > − . Then for every polynomial h with deg( h ) ≤ n , α + 1 , β + 1) Z h ( t ) R ( α,β ) n ( t ) (1 − t ) α t β d t = h (0) . (23) Proof. The difference h ( t ) − h (0) divides by t . Denote by q ( t ) the corresponding quotient.So, q is a polynomial of degree deg( q ) ≤ n − h ( t ) = h (0) + tq ( t ). Then1B( α + 1 , β + 1) Z h ( t ) R ( α,β ) n ( t ) (1 − t ) α t β d t = ( − n h (0)B( α + n + 1 , β + 1) Z Q ( α,β +1) n ( t ) (1 − t ) α t β d t + ( − n B( α + n + 1 , β + 1) Z q ( t ) Q ( α,β +1) n ( t ) (1 − t ) α t β +1 d t. By (17) and (16), the first summand is h (0) and the second is 0.6s a particular case of (23), for β = 0 and m ≤ n ,1 α + 1 Z t m R ( α, n ( t )(1 − t ) α d t = δ m, . (24)Formula (24) was proven in [14] in other way. L ( D , µ α ) For each p, q ∈ N , denote by m p,q the monomial function m p,q ( z ) := z p z q . The inner product of two monomial functions is h m p,q , m j,k i = ( α + 1) B ( p + j + 1 , α + 1) δ p − q,j − k . (25)In particular, this means that the family ( m p,q ) p,q ∈ N is not orthogonal.In this section, we recall various equivalent formulas for an orthonormal basis in L ( D , µ α ), that can be obtained by orthonormalizing ( m p,q ) p,q ∈ N , and whose elementsare known as Jacobi polynomials in z and z , see Koornwinder [21], or disk polynomials ,see W¨unsche [43], among others. These polynomials, in the unweighted case, were alsorediscovered in [23], [29], and [27], in the context of polyanalytic functions. We work witha normalized version of the disk polynomials and define them by b ( α ) p,q ( z ) := ( − p + q e c ( α ) p,q (1 − zz ) − α ∂ q ∂z q ∂ p ∂z p (cid:16) (1 − z z ) p + q + α (cid:17) , (26)where e c ( α ) p,q = s ( α + p + q + 1)Γ( α + p + 1)Γ( α + q + 1)( α + 1) p ! q ! Γ( α + p + q + 1) . (27)Since ∂∂z (1 − zz ) = − z and ∂∂z (1 − zz ) = − z , the expression in (26) can be rewritten inother equivalent forms: b ( α ) p,q ( z ) = ( − q s ( α + p + q + 1)Γ( α + p + 1)( α + 1) p ! q ! Γ( α + q + 1) (1 − zz ) − α ∂ q ∂z q (cid:16) z p (1 − z z ) α + q (cid:17) , (28) b ( α ) p,q ( z ) = ( − p s ( α + p + q + 1)Γ( α + q + 1)( α + 1) p ! q ! Γ( α + p + 1) (1 − zz ) − α ∂ p ∂z p (cid:16) z q (1 − z z ) α + p (cid:17) . (29)By (18), b ( α ) p,q can be expressed via the shifted Jacobi polynomials: b ( α ) p,q ( z ) = s ( α + p + q + 1)Γ( α + p + 1) q !( α + 1) Γ( α + q + 1) p ! z p − q Q ( α,p − q ) q ( | z | ) , if p ≥ q ; s ( α + p + q + 1)Γ( α + q + 1) p !( α + 1) Γ( α + p + 1) q ! z q − p Q ( α,q − p ) p ( | z | ) , if p < q. (30)7he two cases in (30) can be joined and written in terms of (19) and (20): b ( α ) p,q ( rτ ) = c ( α, | p − q | )min { p,q } √ α + 1 r | p − q | τ p − q Q ( α, | p − q | )min { p,q } ( r ) ( r ≥ , τ ∈ T ) , (31) b ( α ) p,q ( rτ ) = τ p − q (1 − r ) − α/ √ α + 1 J ( α, | p − q | )min { p,q } ( r ) . (32)Notice that c ( α, | p − q | )min { p,q } = s ( α + p + q + 1)(min { p, q } )! Γ( α + max { p, q } + 1)(max { p, q } )! Γ( α + min { p, q } + 1) . The family ( b ( α ) p,q ) p,q ∈ N has the following conjugate symmetric property: b ( α ) p,q ( z ) = b ( α ) q,p ( z ) . (33)Applying (14) in the right-hand side of (30) we obtain b ( α ) p,q ( z ) = s ( α + p + q + 1) p ! q !( α + 1) Γ( α + p + 1)Γ( α + q + 1) ×× min { p,q } X k =0 ( − k Γ( α + p + q + 1 − k ) k ! ( p − k )! ( q − k )! z p − k z q − k , (34)In particular, (34) implies that b ( α ) p,q is a polynomial in z and z whose leading term is apositive multiple of the monomials m p − k,q − k .Let P be the set of all polynomials functions in z and z , i.e., the linear span of themonomials P := span { m p,q : p, q ∈ N } . For every ξ ∈ Z and every s ∈ N , denote by W ( α ) ξ,s the subspace of P generated by m p,q with p − q = ξ and min { p, q } < s : W ( α ) ξ,s := span { m p,q : p − q = ξ, min { p, q } < s } . (35)The vector space W ( α ) ξ,s does not depend on α , but we endow it with the inner productfrom L ( D , µ α ). Obviously, dim( W ( α ) ξ,s ) = s . Let us show that W ( α ) ξ,s = span { b ( α ) p,q : p − q = ξ, min { p, q } < s } . (36)Indeed, by (34), m p,q = s ( α + 1)Γ( α + p + 1)Γ( α + q + 1) p ! q !Γ( α + p + q + 2) b ( α ) p,q − p ! q !Γ( α + p + q + 1) min { p,q } X ν =1 ( − ν Γ( α + p + q + 1 − ν ) ν ! ( p − ν )! ( q − ν )! m p − ν,q − ν . (37)8roceeding by induction on s , we see that the monomials m p,q are linear combinations of b ( α ) p − s,q − s with 0 ≤ s ≤ min { p, q } . So, formula (36) means that the first s elements in thediagonal ξ of the table ( b ( α ) p,q ) ∞ p,q =0 generate the same subspace as the first s elements ofthe diagonal ξ in the table ( m p,q ) ∞ p,q =0 . For example, W ( α ) − , = span { m , , m , , m , } = span { b ( α )0 , , b ( α )1 , , b ( α )2 , } , W ( α )1 , = span { m , , m , , m , , m , } = span { b ( α )1 , , b ( α )2 , , b ( α )3 , , b ( α )4 , } . In the following tables we show generators of W ( α )1 , (light blue) and W ( α ) − , (pink). m , m , m , m , m , m , . . . m , m , m , m , m , m , . . . m , m , m , m , m , m , . . . m , m , m , m , m , m , . . . m , m , m , m , m , m , . . . m , m , m , m , m , m , . . .. . . . . . . . . . . . . . . . . . . . . b ( α )0 , b ( α )0 , b ( α )0 , b ( α )0 , b ( α )0 , b ( α )0 , . . . b ( α )1 , b ( α )1 , b ( α )1 , b ( α )1 , b ( α )1 , b ( α )1 , . . . b ( α )2 , b ( α )2 , b ( α )2 , b ( α )2 , b ( α )2 , b ( α )2 , . . . b ( α )3 , b ( α )3 , b ( α )3 , b ( α )3 , b ( α )3 , b ( α )3 , . . . b ( α )4 , b ( α )4 , b ( α )4 , b ( α )4 , b ( α )4 , b ( α )4 , . . . b ( α )5 , b ( α )5 , b ( α )5 , b ( α )5 , b ( α )5 , b ( α )5 , . . .. . . . . . . . . . . . . . . . . . . . .As a consequence, P = S ξ ∈ Z S s ∈ N W ( α ) ξ,s = span { b ( α ) p,q : p, q ∈ N } . Proposition 3.1. The family ( b ( α ) p,q ) p,q ∈ N is an orthonormal basis of L ( D , µ α ) .Proof. The orthonormal property follows straightforwardly from (32) and (21): h b ( α ) p,q , b ( α ) j,k i = 12 π Z π e i( p − q − j + k ) θ d θ Z J ( α, | p − q | )min { p,q } ( t ) J ( α, | j − k | )min { j,k } ( t ) d t = δ p − q,j − k · δ min { p,q } , min { j,k } = δ p,j · δ q,k . By the Stone–Weierstrass theorem, P is dense in C (clos( D )). In turn, by Luzin’s theorem,the set C (clos( D )) | D is dense in L ( D , µ α ) and for every f ∈ C (clos( D )) we have k f k ≤ max z ∈ clos( D ) | f ( z ) | . Now it is easy to see that the set P is dense in L ( D , µ α ), that is, theset of all linear combinations of elements of the family is dense in L ( D , µ α ). For thatreason, ( b ( α ) p,q ) p,q ∈ N is a complete orthonormal family. Corollary 3.2. Let ξ ∈ Z and s ∈ N . Then ( b ( α ) q + ξ,q ) max { s − ,s − ξ − } q =max { , − ξ } is an orthonormal basisof W ( α ) ξ,s . Remark 3.3. By Proposition 3.1 and formula (37), h m ξ + k,k , b ( α ) ξ + q,q i = q ( α +1)Γ( α + p +1)Γ( α + q +1) p ! q !Γ( α + p + q +2)Γ( α + p + q +1) , k = q ;0 , max { , − ξ } ≤ k < q. (38)9he table of basic functions can be expressed as follows: b ( α )0 , ( z ) = h ( α )0 , ( | z | ) , b ( α )0 , ( z ) = z h ( α )0 , ( | z | ) , b ( α )0 , ( z ) = z h ( α )0 , ( | z | ) ,b ( α )1 , ( z ) = z h ( α )1 , ( | z | ) , b ( α )1 , ( z ) = h ( α )1 , ( | z | ) , b ( α )1 , ( z ) = z h ( α )1 , ( | z | ) ,b ( α )2 , ( z ) = z h ( α )2 , ( | z | ) , b ( α )2 , ( z ) = z h ( α )2 , ( | z | ) , b ( α )2 , ( z ) = h ( α )2 , ( | z | ) , where h ( α ) p,q ( t ) := c ( α, | p − q | )min { p,q } √ α +1 Q ( α, | p − q | )min { p,q } ( t ). Below we show explicitly some elements of thisbasis. b ( α )0 , ( z ) = 1 , b ( α )1 , ( z ) = √ α + 2 z, b ( α )2 , ( z ) = r ( α + 3)( α + 2)2 z ,b ( α )1 , ( z ) = r α + 3 α + 1 (cid:0) ( α + 2) zz − (cid:1) , b ( α )2 , ( z ) = r α + 3)( α + 2) α + 1 (cid:18) α + 32 z z − z (cid:19) ,b ( α )2 , ( z ) = r α + 5 α + 1 (cid:18) ( α + 4)( α + 3)2 z z − α + 3) zz + 12 (cid:19) . Now, for every ξ in Z we introduce the subspace W ( α ) ξ associated to the “frequency” ξ or,equivalently, to the diagonal ξ in the tables ( m p,q ) p,q ∈ N and ( b ( α ) p,q ) p,q ∈ Z : W ( α ) ξ := clos(span { m p,q : p − q = ξ } ) = clos [ s ∈ N W ( α ) ξ,s ! . Corollary 3.4. The sequence ( b ( α ) ξ + q,q ) ∞ q =max { , − ξ } is an orthonormal basis of W ( α ) ξ . The space W ( α ) ξ can be naturally identified with L over (0 , , 1) withvarious weights. Proposition 3.5. Each one of the following linear operators is an isometric isomorphismof Hilbert spaces:1) L ((0 , , ( α + 1) (1 − t ) α d t ) → W ( α ) ξ , h f , f ( rτ ) := τ ξ h ( r ) , i.e., f ( z ) := sgn ξ ( z ) h ( zz ) , (39) where z ∈ D , ≤ r < , τ ∈ T ;2) L ((0 , , ( α + 1) t | ξ | (1 − t ) α ) → W ( α ) ξ , h f , f ( rτ ) := τ ξ r | ξ | h ( r ) , i.e., f ( z ) := ( z ξ h ( zz ) , ξ ≥ ,z ξ h ( zz ) , ξ < 0; (40)10 ) L ((0 , → W ( α ) ξ , h f , f ( rτ ) := τ ξ (1 − r ) − α/ √ α + 1 h ( r ) , i.e., f ( z ) := sgn ξ ( z ) (1 − zz ) − α/ √ α + 1 h ( zz ) . (41) Proof. In each case, the isometric property is verified directly using polar coordinates,and the surjective property is justified with the help of the orthonormal basis of W ( α ) ξ (Corollary 3.4). The function sgn : C → C is defined by sgn( z ) := z/ | z | for z = 0 andsgn(0) := 0. Corollary 3.6. The space L ( D , d µ α ) is the orthogonal sum of the subspaces W ( α ) ξ : L ( D , d µ α ) = M ξ ∈ Z W ( α ) ξ . (42)The result of Corollary 3.6 can be seen as the Fourier decomposition of the space L ( D , d µ α ), and each space W ( α ) ξ corresponds to the frequency ξ .Here we show the generators of W ( α )0 (pink) and W ( α ) − (light blue): m , m , m , m , . . . m , m , m , m , . . . m , m , m , m , . . . m , m , m , m , . . .. . . . . . . . . . . . . . . b ( α )0 , b ( α )0 , b ( α )0 , b ( α )0 , . . . b ( α )1 , b ( α )1 , b ( α )1 , b ( α )1 , . . . b ( α )2 , b ( α )2 , b ( α )2 , b ( α )2 , . . . b ( α )3 , b ( α )3 , b ( α )3 , b ( α )3 , . . .. . . . . . . . . . . . . . . It is known [4, Section 1.1] that any n -analytic function can be expressed as a “polynomial”of degree n − z with 1-analytic coefficients, that is, for any f ∈ A n ( D ),there exist analytic functions g , g , . . . , g n − in D such that f ( z ) = n − X k =0 g k ( z ) z k ( z ∈ D ) . Replacing every g k by its Taylor series, we get another classic form of n -analytic functions:there exist coeficients λ j,k in C such that f ( z ) = n − X k =0 ∞ X j =0 λ j,k z j z k ( z ∈ D ) . (43)The following weighted mean value property was proved by Hachadi y Youssfi [14]using a slightly different method. The mean value property for solutions of more generalelliptic equations is studied in [39]. 11 roposition 4.1. Let f ∈ A n ( D ) such that Z D | f ( z ) | (1 − | z | ) α d µ ( z ) < + ∞ . Then f (0) = α + 1 π Z D f ( z ) R ( α, n − ( | z | ) (1 − | z | ) α d µ ( z ) . (44) Proof. For 0 < r ≤ 1, consider I ( r ) := α + 1 π Z r D f ( w ) R ( α, n − ( | w | ) (1 − | w | ) α d µ ( w ) . Then, combining (43) with the polar decomposition, we get that I ( r ) = n − X k =0 ∞ X j =0 λ j,k (cid:18) ( α + 1) Z r u j + k R ( α, n − ( u )(1 − u ) α u d u (cid:19) (cid:18) π Z π e i ( j − k ) θ d θ (cid:19) . The terms in the inner series vanish whenever j = k . Setting t instead of u in the integral,we have I ( r ) = n − X k =0 λ k,k ( α + 1) Z r t k R ( α, n − ( t )(1 − t ) α d t. Take limits in both sides when r tends to 1. Apply Lebesgue’s dominated convergencetheorem to get the integral over D in the left-hand side and the integral over [0 , 1) in theright-hand side. By formula (24), I (1) = n − X k =0 λ k,k ( α + 1) Z t k R ( α, n − ( t )(1 − t ) α d t = n − X k =0 λ k,k δ k, = λ , = f (0) . For α = 0, Proposition 4.1 reduces to the following mean value property that appearedin [23] and [27]. Corollary 4.2. Let z ∈ C , r > , and f ∈ A n ( z + r D ) such that Z z + r D | f ( w ) | d µ ( w ) < + ∞ . Then f ( z ) = α + 1 π r Z z + r D f ( w ) R ( α, n − (cid:18) | w − z | r (cid:19) d µ ( w ) . (45) Proof. Denote by ϕ the linear change of variables ϕ ( w ) := rw + z . If f ∈ A n ( z + r D ),then f ◦ ϕ ∈ A n ( D ). Applying (44) to f ◦ ϕ , we obtain (45).12 eighted Bergman spaces of polyanalytic functions on general complexdomains Given n in N , an open subset Ω of C and a continuous function W : Ω → (0 , + ∞ ),we denote by A n (Ω , W ) the space of n -analytic functions belonging to L (Ω , W ) andprovided with the norm of L (Ω , W ). The mean value property (45) implies that theevaluation functionals in A n (Ω , W ) are bounded (moreover, they are uniformly boundedon compacts), and A n (Ω , W ) is a RKHS. Here are proofs of these facts. Lemma 4.3. Let K be a compact subset of Ω . Then there exists a number C n,W,K > such that for every f in A n (Ω , W ) and every z in K , | f ( z ) | ≤ C n,W,K k f k A n (Ω ,W ) . (46) Proof. Let r be the distance from K to C \ Ω. Since K is compact and C \ Ω is closed, r > 0. Put r := min { r / , } , K := { w ∈ C : d ( w, K ) ≤ r } , C := (cid:18) max ≤ t ≤ | R ( α, n − ( t ) | (cid:19) max w ∈ K p W ( w ) ! . For every z in K , we estimate | f ( z ) | from above applying (45) and Schwarz inequality: | f ( z ) | ≤ πr Z z + r D | f ( w ) | (cid:12)(cid:12)(cid:12)(cid:12) R ( α, n − (cid:18) | w − z | r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) d µ ( w ) ≤ C πr Z z + r D | f ( w ) | p W ( w ) d µ ( w ) ≤ C πr (cid:18)Z z + r D | f ( w ) | W ( w ) d µ ( w ) (cid:19) / (cid:18)Z z + r D µ ( w ) (cid:19) / ≤ C √ πr k f k A n (Ω ,W ) . So, (46) is fulfilled with C n,W,K = C √ πr . Proposition 4.4. A n (Ω , W ) is a RKHS.Proof. Given a Cauchy sequence in A n (Ω , W ), for every compact K it converges uniformlyon K by Lemma 4.3. The pointwise limit of this sequence is also polyanalytic by [4,Corollary 1.8], and it coincides a.e. with the limit in L (Ω , W ). Lemma 4.3 also assuresthe boundedness of the evalution functionals and thereby the existence of the reproducingkernel. See similar proofs in [26, Proposition 3.3].Denote by A n ) (Ω , W ) the orthogonal complement of A n − (Ω , W ) in A n (Ω , W ). Corollary 4.5. A n ) (Ω , W ) is a RKHS. Weighted Bergman spaces of polyanalytic functions onthe unit disk In the rest of the paper, we suppose that n ∈ N and α > − 1. Given z in D , denote by K ( α ) n,z the reproducing kernel of A n ( D , µ α ) at the point z and by K ( α )( n ) ,z the reproducingkernel of A n ) ( D , µ α ) at the point z . Hachadi and Youssfi [14] computed the reproducingkernel of A n ( D , µ α ): K ( α ) n,z ( w ) = (1 − wz ) n − (1 − zw ) n +1 R ( α, n − (cid:12)(cid:12)(cid:12)(cid:12) z − w − zw (cid:12)(cid:12)(cid:12)(cid:12) ! . (47)Their method uses (44) and a generalization of the unitary operator constructed by Pes-soa [27]. Formula (47) implies an exact expression for the norm of K ( α ) n,z , which is also thenorm of the evaluation functional at the point z : k K ( α ) n,z k = s ( n + α ) (cid:18) n + α − n − (cid:19) − | z | . (48)Obviously, the reproducing kernel of A n ) ( D , µ α ) can be written as K ( α )( n ) ,z ( w ) = K ( α ) n,z ( w ) − K ( α ) n − ,z ( w ) . (49)Unfortunately, we are unable to obtain a simpler formula for K ( α )( n ) ,z . Orthonormal basis in A n ( D , µ α ) Proposition 5.1. The family ( b ( α ) p,q ) p ∈ N ,q 0, using expansion (43) and the orthogonality of the Fourier basis on T , we easilyobtain Z r D f b ( α ) p,q d µ α = n − X k =0 λ k + p − q,k Z r D m k + p − q,k b ( α ) p,q d µ α . The dominated convergence theorem allows us to pass to integrals over D , because f b ( α ) p,q and m k + p − q,k b ( α ) p,q belong to L ( D , µ α ). Now the assumption f ⊥ b ( α ) p,q = 0 yields n − X k =0 h m k + p − q,k , b ( α ) p,q i λ k + p − q,k = 0 ( p ∈ N , ≤ q < n ) . (50)14or a fixed ξ in Z with ξ > − n , put s = min { n, n + ξ } . The vector [ λ k + ξ,k ] n − k =max { , − ξ } satisfies the homogeneous linear system (50) with the s × s matrix h h m ξ + k,k , b ( α ) ξ + q,q i i n − q,k =max { , − ξ } . By (38), this is an upper triangular matrix with nonzero diagonal entries, hence the uniquesolution of (50) is zero. Corollary 5.2. The sequence ( b ( α ) p,n − ) p ∈ N is an orthonormal basis of A n ) ( D , µ α ) . We denote by P ( α ) n and P ( α )( n ) the orthogonal projections acting in L ( D , µ α ), whoseimages are A n ( D , µ α ) and A n ) ( D , µ α ), respectively. They can be computed in terms ofthe corresponding reproducing kernels:( P ( α ) n f )( z ) = h f, K ( α ) n,z i , ( P ( α )( n ) f )( z ) = h f, K ( α )( n ) ,z i . For example, ( b ( α ) p,q ) p ∈ N ,q< is an orthonormal basis of A ( D , µ α ), and ( b ( α ) p, ) p ∈ N is anorthonormal basis of A ( D , µ α ). b ( α )0 , b ( α )0 , b ( α )0 , b ( α )0 , b ( α )0 , . . .b ( α )1 , b ( α )1 , b ( α )1 , b ( α )1 , b ( α )1 , . . .b ( α )2 , b ( α )2 , b ( α )2 , b ( α )2 , b ( α )2 , . . .b ( α )3 , b ( α )3 , b ( α )3 , b ( α )3 , b ( α )3 , . . . ... ... ... ... ... . . . b ( α )0 , b ( α )0 , b ( α )0 , b ( α )0 , b ( α )0 , . . .b ( α )1 , b ( α )1 , b ( α )1 , b ( α )1 , b ( α )1 , . . .b ( α )2 , b ( α )2 , b ( α )2 , b ( α )2 , b ( α )2 , . . .b ( α )3 , b ( α )3 , b ( α )3 , b ( α )3 , b ( α )3 , . . . ... ... ... ... ... . . . Decomposition of A n ( D , µ α ) into subspaces corresponding to different“frequences” We will use the following elementary fact about orthonormal bases in Hilbert spaces. Proposition 5.3. Let H be a Hilbert space and B ⊆ H be an orthonormal basis of H (in this proposition we treat orthonormal bases like sets rather than families). Supposethat B and B are some subsets of B . Denote by H and H the closed subspaces of H generated by B and B , respectively. Then B ∩ B is an orthonormal basis of H ∩ H . Applying Proposition 5.3 to the Hilbert space L ( D , µ α ) and thinking in terms oforthonormal bases (see Propositions 3.1, 5.1, and Corollaries 3.2, 3.4), we easily find theintersection of W ( α ) ξ and A n ( D , µ α ): W ( α ) ξ ∩ A n ( D , µ α ) = ( W ( α ) ξ, min { n,n + ξ } , ξ ≥ − n + 1; { } , ξ < − n + 1 . (51)Here is a description of the subspaces W ( α ) ξ,m in terms of the polar coordinates.15 roposition 5.4. For every ξ in Z and every s in N , the space W ( α ) ξ,s consists of allfunctions of the form f ( rτ ) = τ ξ r | ξ | Q ( r ) ( r ≥ , τ ∈ T ) , where Q is a polynomial of degree ≤ s − . Moreover, k f k = k Q k L ([0 , , ( α +1)(1 − t ) α t | ξ | d t ) . Proof. The result follows directly by Proposition (3.5) and formula (40).The decomposition of A n ( D , µ α ) into a direct sum of the “truncated frequency sub-spaces” shown below follows from Proposition 5.1 and Corollary 3.2, and plays a crucialrole in the study of radial operators. It can be seen as the “Fourier series decomposition”of A n ( D , µ α ). Proposition 5.5. A n ( D , µ α ) = ∞ M ξ = − n +1 W ( α ) ξ, min { n,n + ξ } . (52)Let us illustrate Proposition 5.5 for n = 3 with a table (we have marked in differentshades of blue the basic functions that generate each truncated diagonal): b ( α )0 , b ( α )0 , b ( α )0 , b ( α )0 , . . .b ( α )1 , b ( α )1 , b ( α )1 , b ( α )1 , . . .b ( α )2 , b ( α )2 , b ( α )2 , b ( α )2 , . . .b ( α )3 , b ( α )3 , b ( α )3 , b ( α )3 , . . . ... ... ... ... . . .Define U ( α ) n : A n ( D , µ α ) → L ∞ ξ = − n +1 C min { n,n + ξ } ,( U ( α ) n f ) ξ,q := h f, b ( α ) q + ξ,q i ( ξ ≥ − n + 1 , max { , − ξ } ≤ q ≤ n − . (53)Here, for − n + 1 ≤ ξ < 0, the componentes of vectors in C n + ξ are enumerated from − ξ to n − Proposition 5.6. The operator U ( α ) n is an isometric isomorphism of Hilbert spaces.Proof. Follows from Proposition 5.1 or, even easier, from Proposition 5.5 and the fact that( b ( α ) q + ξ,q ) n − q =max { , − ξ } is an orthonormal basis of W ( α ) ξ, min { n,n + ξ } (see Corollary 3.2).An analog of the upcoming fact for the unweighted poly-Bergman space was provedby Vasilevski [42, Section 4.2]. We obtain it as a corollary from Proposition 3.1 andCorollary 5.2. Corollary 5.7. The space L ( D , µ α ) is the orthogonal sum of the subspaces A m ) ( D , µ α ) , m ∈ N : L ( D , µ α ) = M m ∈ N A m ) ( D , µ α ) . The set of Toeplitz operators is not weakly dense Given a Hilbert space H , we denote by B ( H ) the algebra of all bounded operators actingin H . If H is a RKHS naturally embedded into L ( D , µ α ) and S ∈ B ( H ), then the Berezintransform of S is defined byBer H ( S )( z ) := h SK z , K z i H h K z , K z i H , i.e. , Ber H ( S )( z ) = ( SK z )( z ) K z ( z ) . The Berezin transform can be considered as a bounded linear operator B ( H ) → L ∞ (Ω).Stroethoff proved [36] that Ber H is injective for various RKHS of analytic functions, inparticular, for H = A ( D ). Engliˇs noticed [10, Section 2] that Ber H is not injectivefor various RKHS of harmonic functions. The idea of Engliˇs can be applied withoutany changes to various spaces of polyanalytic and polyharmonic functions. For clarity ofpresentation, we state the result of Engliˇs for A n ( D , µ α ), n ≥ 2, and repeat his proof. Proposition 6.1. Let H = A n ( D , µ α ) with n ≥ . Then the Berezin transform Ber H isnot injective.Proof. Let f ∈ H such that f ∈ H and the functions f, f are linearly independent. Forexample, f ( z ) := z . Following the idea from [10, Section 2], consider the operator Sh := h h, f i H f − h h, f i H f . Then S = 0, but h SK z , K z i H = | f ( z ) | − | f ( z ) | = 0 for every z in D . So, Ber H ( S ) is thezero constant.Given a function g in L ∞ ( D ), let M g be the multiplication operator defined on L ( D , µ α )by M g f := gf . If H is a closed subspace of L ( D , µ α ), then the Toeplitz operator T H,g isdefined on H by T H,g ( f ) := P H ( gf ) = P H M g f. For H = A n ( D , µ α ) and H = A n ) ( D , µ α ), we write just T ( α ) n,g and T ( α )( n ) ,g , respectively. Theproof of the following fact is the same as the proof of [26, Proposition 3.18] or the proofof [7, Theorem 4]. Proposition 6.2. If g ∈ L ∞ ( D ) and T ( α ) n,g = 0 , then g = 0 a.e. In other words, thefunction g T ( α ) n,g , acting from L ∞ ( D ) to B ( A n ( D , µ α )) , is injective. Inspired by the idea of Engliˇs explained in the proof of Proposition 6.1, we will provethat set of Toeplitz operators is not weakly dense in B ( A n ( D , µ α )) with n ≥ 2. First, letus prove an auxiliary fact from linear algebra: bounded quadratic forms separate linearlyindependent vectors. Lemma 6.3. Let H be a Hilbert space and f, g be two linearly independent vectors in H .Then there exists S in B ( H ) such that h Sf, f i H = h Sg, g i H . roof. Without lost of generality, we will suppose that k f k H = 1. Decompose g into thelinear combination g = λ f + λ h , with λ , λ ∈ C , k h k H = 1, h ⊥ f . More explicitly, λ := h g, f i H , w := g − λ f, λ := k w k H , h := 1 λ w. Define S as the orthogonal projection onto h : Sv := h v, h i H h ( v ∈ H ) . Then Sf = 0 and Sg = λ h , therefore h Sf, f i H = 0 and h Sg, g i H = λ > Theorem 6.4. Let H = A n ( D , µ α ) with n ≥ . Then the set of the Toeplitz operatorswith bounded symbols is not weakly dense in B ( H ) .Proof. Let f ∈ H such that f ∈ H and the functions f, f are linearly independent. Forexample, f ( z ) := z . The set W := { S ∈ B ( H ) : h Sf, f i H = h Sf , f i H } is a weakly closed subspace of B ( H ). By Lemma 6.3, W = B ( H ). On the other hand, forevery a in L ∞ ( D ) h T ( α ) n,a f, f i H = Z X a | f | d µ α = h T ( α ) n,a f , f i H , i.e., { T ( α ) n,a : a ∈ L ∞ ( D ) } ⊆ W . Remark 6.5. An analog of Theorem 6.4 is true for the space of µ α -square integrable n -harmonic functions on D , with n ≥ Set of operators diagonalized by a family of subspaces The theory of von Neumann algebras and their decompositions is well developed. For ourpurposes, it is sufficient to use the following elementary scheme from [26]. This scheme issimilar to ideas from [12, 28, 46]. Definition 7.1. Let H be a Hilbert space, U be a self-adjoint subset of B ( H ), and ( W j ) j ∈ J be a finite or countable family of nonzero closed subspaces of H such that H = L j ∈ J W j .We say that this family diagonalizes U if the following two conditions are satisfied.1. For each j in J and each U in U , there exists λ U,j in C such that W j ⊆ ker( λ U,j I − U ),i.e., U ( v ) = λ U,j v for every v in W j .2. For every j , k in J with j = k , there exists U in U such that λ U,j = λ U,k .18 roposition 7.2. Let H , U , and ( W j ) j ∈ J be like in Definition 7.1. Denote by A thecommutant of U . Then A consists of all bounded linear operators that act invariantly oneach of the subspaces W j , with j ∈ J : A = { S ∈ B ( H ) : ∀ j ∈ J S ( W j ) ⊆ W j } . (54) Furthermore, A is isometrically isomorphic to L j ∈ J B ( W j ) , and the von Neumann algebragenerated by U is isometrically isomorphic to L j ∈ J C I W j . Example 7.3. Let j , . . . , j m ∈ J , λ , . . . , λ m ∈ C , and u j k , v j k ∈ W j k for every k in { , . . . , m } . Then the operator S : H → H defined by Sf := m X k =1 λ k h f, u j k i v j k , (55)belongs to A . Moreover, every operator of finite rank, belonging to A , can be written inthis form. See the proof of [26, Corollary 5.7] for a similar situation. Proposition 7.4. Let H , U , and ( W j ) j ∈ J be like in Definition 7.1, and H be a closedsubspace of H invariant under U . For every U in U , denote by U | H H the compression of U onto the invariant subspace H , and put U := n U | H H : U ∈ U o , J := { j ∈ J : W j ∩ H = { }} . Then H = M j ∈ J ( W j ∩ H ) , (56) and the family ( W j ∩ H ) j ∈ J diagonalizes U . Example 7.5. The operators of finite rank, commuting with U | H H for every U in U , areof the form (55), but with u j k , v j k ∈ W j k ∩ H . Radial operators in L ( D , µ α ) For each τ in T , we denote by ρ ( α ) ( τ ) the rotation operator acting in L ( D , µ α ) by therule ( ρ ( α ) ( τ ) f )( z ) := f ( τ − z ) . (57)It is easy to see that ρ ( α ) ( τ τ ) = ρ ( α ) ( τ ) ρ ( α ) ( τ ), the operators ρ ( α ) ( τ ) are unitary, andfor every f in L ( D , µ α ) the mapping τ ρ ( α ) ( τ ) f is continuous (this is easy to check firstfor the case when f is a continuous function with compact support). So, ( ρ ( α ) , L ( D , µ α ))is a unitary representation of the group T . The operators commuting with ρ ( α ) ( τ ) for every τ in T are called radial operators . We denote the set of all radial operators in L ( D , µ α )by R ( α ) : R ( α ) := { ρ ( α ) ( τ ) : τ ∈ T } ′ = { S ∈ B ( L ( D , µ α )) : ∀ τ ∈ T ρ ( α ) ( τ ) S = Sρ ( α ) ( τ ) } . Since { ρ ( α ) ( τ ) : τ ∈ T } is an autoadjoint subset of B ( L ( D , µ α )), its commutant R ( α ) is avon Neumann algebra [34].Recall that the subspaces W ( α ) ξ are defined by (35).19 emma 7.6. The family ( W ( α ) ξ ) ξ ∈ Z diagonalizes the collection { ρ ( α ) ( τ ) : τ ∈ T } in thesense of Definition 7.1.Proof. 1. Let τ ∈ T . For every p, q ∈ Z with p − q = ξ , formula (31) implies ρ ( α ) ( τ ) b ( α ) p,q = τ q − p b ( α ) p,q = τ − ξ b ( α ) p,q , (58)i.e., b ( α ) p,q ∈ ker( τ − ξ I − ρ ( α ) ( τ )). By Corollary 3.4, the functions b ( α ) p,q with p − q = ξ forman orthonormal basis of W ( α ) ξ . So, W ( α ) ξ ⊆ ker( τ − ξ I − ρ ( α ) ( τ )) . (59)2. Let ξ , ξ ∈ Z and ξ = ξ . Put τ = exp i πξ − ξ . Then τ − ξ = τ − ξ . Proposition 7.7. The von Neumann algebra R ( α ) consists of all operators that act in-variantly on W ( α ) ξ for every ξ in Z , and is isometrically isomorphic to L ξ ∈ Z B ( W ( α ) ξ ) .Proof. Follows from Proposition 7.2 and Lemma 7.6.The radialization transform Rad ( α ) : B ( L ( D , µ α )) → B ( L ( D , µ α )), introduced by Zor-boska [46], acts by the ruleRad ( α ) ( S ) := Z T ρ ( τ ) Sρ ( τ − ) d µ T ( τ ) , where µ T is the normalized Haar measure on T , and the integral is understood in the weaksense. The condition S ∈ R ( α ) is equivalent to Rad ( α ) ( S ) = S . Radial operators in A n ( D , µ α ) Proposition 7.8. The space A n ( D , µ α ) is invariant under every rotation ρ ( α ) ( τ ) , τ ∈ T .First proof. The reproducing kernel of A n ( D , µ α ), given by (47), is invariant under simul-taneous rotations in both arguments: K ( α ) n,τz ( τ w ) = K ( α ) n,z ( w ) ( z, w ∈ D , τ ∈ T ) . (60)According to [26, Proposition 4], this implies the invariance of the subspace. Second proof. By (57), the elements of the orthonormal basis ( b ( α ) p,q ) p ∈ N , ≤ q 0, and sup ξ ≥− n +1 k A ξ k < + ∞ . Being a direct sum of W*-algebras, M n is a W*-algebra. We identify the elements of M n with the bounded linear operators acting in L ∞ ξ = − n +1 C min { n,n + ξ } . Now we are ready todescribe the structure of R ( α ) n . Recall that U ( α ) n is given by (53). Theorem 7.9. Let n ∈ N . Then R ( α ) n consists of all operators belonging to B ( A n ( D , µ α )) that act invariantly on each subspace W ( α ) ξ, min { n,n + ξ } , for ξ ≥ − n + 1 . Furthermore, R ( α ) n ∼ = ∞ M ξ = − n +1 B ( W ( α ) ξ, min { n,n + ξ } ) , and R ( α ) n is spatially isomorphic to M n : U ( α ) n R ( α ) n ( U ( α ) n ) ∗ = M n . Proof. We apply the scheme from Propositions 7.2, 7.4, W j = W ( α ) ξ , U = { ρ ( α ) ( τ ) : τ ∈ T } , and H = A n ( D , µ α ). By (51), we obtain J = { ξ ∈ Z : ξ ≥ − n + 1 } , A n ( D , µ α ) ∩ W ( α ) ξ = W ( α ) ξ, min { n,n + ξ } . So, the W*-algebra R ( α ) n is isometrically isomorphic to the direct sum of B ( W ( α ) ξ, min { n,n + ξ } ),with ξ ≥ − n + 1. Using the orthonormal basis ( b ( α ) ξ + k,k ) n − k =max { , − ξ } of W ( α ) ξ, min { n,n + ξ } , werepresent linear operators on this space as matrices. Define Φ ( α ) n : R ( α ) n → M n byΦ ( α ) n ( S ) := (cid:18)hD Sb ( α ) ξ + k,k , b ( α ) ξ + j,j Ei n − j,k =max { , − ξ } (cid:19) ∞ ξ = − n +1 . (61)In other words, Φ ( α ) n ( S ) = U ( α ) n S ( U ( α ) n ) ∗ , i.e., Φ ( α ) n is an isometrical isomorphism of W*-algebras induced by the unitary operator U ( α ) n .21adial operators of finite rank, acting in A n ( D , µ α ), can be constructed as in Exam-ples 7.3 and 7.5.It is easy to verify (see a more general result in [26, Corollary 4.3]) that if A n = A n ( D , µ α ) and S ∈ R ( α ) n , then Ber A n ( S ) is a radial function. For n = 1, the Berezintransform Ber A is injective. So, if S ∈ B ( A ( D , µ α )) and the function Ber A ( S ) is radial,then the operator S is radial. Radial operators in A n ) ( D , µ α ) Let n ∈ N . The space A n ) ( D , µ α ) is invariant under the rotation ρ ( α ) ( τ ) for all τ in T .The proof is similar to the proof of Proposition 7.8. Denote the compression of ρ ( α ) ( τ )onto A n ) ( D , µ α ) by ρ ( n ) ( τ ). Let R ( α )( n ) be the von Neumann algebra of all radial operatorsin A n ) ( D , µ α ). Theorem 7.10. R ( α )( n ) consists of all operators belonging to B ( A n ) ( D , µ α )) that are diag-onal with respect to the orthonormal basis ( b ( α ) p,n − ) ∞ p =0 . Furthermore, R ( α )( n ) ∼ = ℓ ∞ ( N ) . Proof. Corollaries 3.4 and 5.2 give W ( α ) ξ ∩ A n ) ( D , µ α ) = ( C b ( α ) ξ + n − ,n − , ξ ≥ − n + 1 , { } , ξ < − n + 1 . (62)By Propositions 7.2, 7.4 and formula (62), R ( α )( n ) consists of the operators that act invari-antly on C b ( α ) ξ + n − ,n − , ξ ≥ − n + 1, i.e., are diagonal with respect to the basis ( b ( α ) p,n − ) ∞ p =0 .Therefore the function Φ ( α )( n ) : R ( α )( n ) → ℓ ∞ ( N ), defined byΦ ( α )( n ) ( S ) = (cid:0) h Sb ( α ) p,n − , b ( α ) p,n − i (cid:1) ∞ p =0 , (63)is an isometric isomorphism. This section is similar to [26, Section 6], but here we use Jacobi polynomials instead ofthe generalized Laguerre polynomials. Radial functions Given g in L ∞ ( D ), define rad( g ) : D → C byrad( g )( z ) := Z T g ( τ z ) d µ T ( τ ) . (64)22iven a in L ∞ ([0 , e a : D → C by e a ( z ) := a ( | z | ) ( z ∈ D ) . The proof of the following criterion is a simple exercise. Proposition 8.1. Given g in L ∞ ( D ) , the following conditions are equivalent:(a) for every τ in T , the equality g ( τ z ) = g ( z ) is true for a.e. z in D ;(b) for every τ in T , the equality ρ ( α ) ( τ ) g = g is true a.e.;(c) rad( g ) = g a.e.;(d) there exists a in L ∞ ([0 , such that g = e a a.e. Radial multiplication operators in L ( D , µ α ) Proposition 8.2. Let g ∈ L ∞ ( D ) . Then Rad ( α ) ( M g ) = M ( α )rad( g ) . Given a in L ∞ ([0 , β a,α,ξ,j,k by β a,α,ξ,j,k := Z a ( √ t ) J ( α, | ξ | )min { j,j + ξ } ( t ) J ( α, | ξ | )min { k,k + ξ } ( t ) d t, (65)i.e., β a,α,ξ,j,k := c ( α, | ξ | )min { q + ξ,q } c ( α, | ξ | )min { k + ξ,k } Z a ( √ t ) Q ( α, | ξ | )min { q,q + ξ } ( t ) Q ( α, | ξ | )min { k,k + ξ } ( t ) (1 − t ) α t | ξ | d t. (66) Proposition 8.3. Let a ∈ L ∞ ([0 , . Then M e a ∈ R ( α ) , and h M e a b ( α ) p,q , b ( α ) j,k i = h e ab ( α ) p,q , b ( α ) j,k i = δ p − q,j − k β a,α,p − q,q,k . (67) Proof. Since e a is invariant under rotations, it follows directly from definitions that M ( α ) e a commutes with ρ ( α ) ( τ ) for every τ . This is a particular case of [26, Lemma 4.4]. For-mula (67) is obtained directly using polar coordinates. Radial Toeplitz operators in A n ( D , µ α ) Proposition 8.4. Let g ∈ L ∞ ( D ) . Then T ( α ) n,g is radial if and only if the function g isradial.Proof. Follows from Proposition 6.2 and [26, Corollaries 4.6, 4.7].Recall that Φ ( α ) n : R ( α ) n → M n is defined by (61).Given a in L ∞ ([0 , γ ( α ) n ( a ) the sequence of matrices [ γ ( α ) n ( a ) ξ ] ∞ ξ = − n +1 ,where γ ( α ) n ( a ) ξ ∈ M min { n + ξ,n } and γ ( α ) n ( a ) ξ := (cid:2) β a,α,ξ,j,k (cid:3) n − j,k =max { , − ξ } . (68)23 roposition 8.5. Let a ∈ L ∞ ([0 , . Then T ( α ) n, e a ∈ R ( α ) n and Φ n ( T ( α ) n, e a ) = γ ( α ) n ( a ) .Proof. Apply Propositions 8.3 and 8.4. Radial Toeplitz operators in A n ) ( D , µ α ) Proposition 8.6. Let a ∈ L ∞ ([0 , . Then T ( α )( n ) , e a ∈ R ( α )( n ) , the operator T ( α )( n ) , e a is diagonalwith respect to the orthonormal basis ( b ( α ) p,n − ) ∞ p =0 , and the corresponding eigenvalues canbe computed by λ a,α,n ( p ) = Z a ( √ t ) (cid:0) J ( α, | p − n +1 | )min { p,n − } ( t ) (cid:1) d t ( p ∈ N ) . (69) Proof. From Proposition 8.4 we get T ( α )( n ) , e a ∈ R ( α )( n ) . Due to Proposition 8.3 and Theo-rem 7.10, λ a,α,n ( p ) = (Φ ( n ) ( T ( α )( n ) , e a )) p = h T ( α )( n ) , e a b ( α ) p,n − , b ( α ) p,n − i = β a,p − n +1 ,n − ,n − . Acknowledgements The research has been supported by CONACYT (Mexico) project “Ciencia de Frontera2019”, N 61517, by IPN-SIP project 20200650 (Instituto Polit´ecnico Nacional, Mexico),and by CONACYT scholarships. Many ideas of this paper were inspired by talks or jointworks with Nikolai Vasilevski, Maribel Loaiza Leyva, Ana Mar´ıa Teller´ıa Romero, IsidroMorales Garc´ıa, and Jorge Iv´an Correo Rosas. 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