HERMITIAN WEIGHTED COMPOSITION OPERATORS AND BERGMAN EXTREMAL FUNCTIONS

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1 HERMITIAN WEIGHTED COMPOSITION OPERATORS AND BERGMAN EXTREMAL FUNCTIONS CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO Abstract. Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are wz) κ for κ, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.. Introduction In this paper, we give necessary conditions Theorem 3) on the symbols f and ϕ for W f,ϕ to be a Hermitian weighted composition operator on a weighted Hardy space. For the standard weight Bergman spaces A 2 α for α 0, all of which are also weighted Hardy spaces, we establish the converse Theorem 6) and identify all of the Hermitian weighted composition operators for these spaces. In Sections 2 and 3, we identify the spectra and spectral decompositions of these operators. Section 3 covers the Hermitian weighted composition operators with continuous spectra and we establish a unitary equivalence Theorem 7) between these operators and multiplication operators on an L 2 [0, ], µ) space. In addition, we show that these operators are part of an analytic semigroup of normal weighted composition operators Corollary 9). In the final section of the paper, we apply the results of Section 3 to find the extremal functions Theorem 25) for the invariant subspaces for multiplication by z in the Bergman spaces A 2 α for α 0 that are Date: July 22, Mathematics Subject Classification. Primary: 47B32; Secondary: 47B33, 47B5, 47B38, 30H20. Key words and phrases. weighted composition operator, composition operator, Hermitian operator, normal operator, weighted Hardy space, weighted Bergman space, spectral measure, Bergman inner function, projected kernel function. Eungil Ko is supported by the Basic Science Research Program through the National Research Foundation of Korea NRF) funded by the Ministry of Education, Science and Technology ). The authors would like to thank Ehwa Women s University, Seoul, and Purdue University, West Lafayette, for their generosity in hosting the authors during their collaboration.

2 2 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO associated with the usual atomic inner functions on the disk. Finally, explicit formulas for the kernel functions for these subspaces are computed Theorem 26). Weighted composition operators have been studied occasionally over the past few decades, but have usually arisen in answering other questions related to operators on spaces of analytic functions, such as questions about multiplication operators or composition operators. For example, Forelli [3] showed that the only isometries of H p for p 2 are weighted composition operators and that the isometries for H p with p 2 have analogues that are isometries of H 2 but there are also many other isometries of H 2 ). Weighted composition operators also arise in the description of commutants of analytic Toeplitz operators see for example [3, 4]) and in the adjoints of composition operators see for example [6, 7, 8, 0]). Only recently have investigators begun to study the properties of weighted composition operators in general for example, see [5]). Our goal in this paper is to characterize the self-adjoint weighted composition operators on a very broad class of spaces, the weighted Hardy spaces. A Hilbert space H whose vectors are functions analytic on the unit disk D is called [9, p. 4] a weighted Hardy space if the polynomials are dense and the monomials, z, z 2,, are an orthogonal set of non-zero vectors in H. Each weighted Hardy space is characterized by a weight sequence, β, defined for each non-negative integer j by βj) z j. For this paper, we will assume that the norm has been scaled so that β0). For a given weight sequence, β, the corresponding weighted Hardy space will be denoted H 2 β), and its inner product is given by a j z j, c j z j a j c j βj) 2 j0 j0 for functions in H 2 β). If f is analytic on the open unit disk, D, and ϕ is an analytic map of the unit disk into itself, the weighted composition operator on H 2 β) with symbols f and ϕ is the operator W f,ϕ h)z) fz)hϕz)) for h in H 2 β). Letting T f denote the analytic multiplication operator given by T f h) fh and C ϕ the composition operator given by C ϕ h) h ϕ for h in H 2 β), if T f and C ϕ are both bounded operators, then clearly W f,ϕ is bounded on H 2 β) and j0 W f,ϕ T f C ϕ T f C ϕ Although it will have little impact on our work, for any weighted Hardy space, if T f is bounded, then f is in H, but it is not necessary for T f to be bounded for W f,ϕ to be bounded see [5]). In an earlier paper, Cowen and Ko [] identified the Hermitian weighted composition operators on the standard Hardy Hilbert space, H 2, that is, in the case βj). Specifically, they showed that if W f,ϕ is Hermitian, then f and ϕ are related linear fractional maps that can be separated into three distinct cases, some for which W f,ϕ is a real multiple of a unitary operator as well as being Hermitian, some for which W f,ϕ is compact, and some for which W f,ϕ has no eigenvalues. In each case, the spectral measures were computed and in some of the cases, further information was provided. Generally speaking, the results for more general spaces H 2 β) divide into the same three cases. Many of the techniques used in that paper

3 HERMITIAN WEIGHTED COMPOSITION OPERATORS 3 carry over to cases studied in this paper, as will be noted in the specific results. In the cases in which the proof in the general weighted Hardy space is essentially the same as in the usual Hardy space, we will omit the proof and refer the reader to []. In this paper, we give necessary conditions Theorem 3) on the symbols f and ϕ for W f,ϕ to be a Hermitian weighted composition operator on H 2 β). We prove the converse of this theorem Theorem 6) for weighted Hardy spaces where the kernels for evaluation of the functions in the space are K w z) wz) κ for κ ; we will write H 2 β κ ) for this weighted Hardy space. In the classical Hardy space, κ and in the Bergman space, κ 2. Spaces with other values of κ are studied in [2], for example, and in [9, p. 27], it is noted that these spaces are equivalent to the standard weight Bergman spaces. In particular, for κ >, the weighted Bergman space A 2 κ 2 consists of the same functions as H2 β κ ) and fz) 2 κ ) z 2 ) κ 2 da π D is another expression for the norm for the space. Specifically, we show that, as in the usual Hardy space, the Hermitian weighted composition operators fall into three classes, the compact weighted composition operators, the multiples of isometries, both covered in Section 2), and those that have no eigenvalues Section 3). In the first two cases, we identify the eigenvalues and eigenvectors in H 2 β κ ), providing spectral resolutions for these operators) and in the third case, for general spaces H 2 β κ ), we see that the Hermitian weighted composition operators are cyclic, are part of an analytic semigroup that includes normal weighted composition operators, and have no eigenvalues. We find measures µ µ κ on [0, ] that allow us to identify the spectral resolutions and a unitary equivalence with multiplication operators {M x t} on L 2 µ). In addition, as a consequence of the calculation of the spectral measures, using properties of the weighted composition operators and the unitary equivalence, we compute the extremal functions associated with the Bergman shift-invariant subspaces associated with the usual atomic inner functions and we find the reproducing kernel functions for these subspaces Theorems 25 and 26). The authors would like to thank Peter Duren, Alexander Schuster, and Kehe Zhu, not only for their books [2] and [7] but also for their help with references to the literature concerning extremal functions for Bergman spaces. We begin by proving a result that allows us to choose standard forms for the operators under study. Proposition. Let f be analytic on the unit disk and let ϕ be an analytic map of the unit disk into itself. For θ a real number, let U θ be the composition operator given by U θ h)z) he iθ z) for h in H 2 β). The operator U θ is unitary on H 2 β), and if W f,ϕ is bounded, then U θ W f,ϕu θ W f, eϕ where fz) fe iθ z) and ϕz) e iθ ϕe iθ z) Proof. The calculation is essentially the same as in the proof of the corresponding result in [], and is omitted.

4 4 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO This proposition will permit us to choose convenient symbols for the weighted composition operators we study without losing any generality of the properties of the operators we are trying to understand. Corollary 2. If f analytic in the unit disk and ϕ an analytic map of the disk into itself determine a bounded weighted composition operator, W f,ϕ, there are g analytic in the disk and ψ an analytic map of the unit disk into itself with ψ0) 0 so that the weighted composition operator W g,ψ is unitarily equivalent to W f,ϕ. Proof. Choose θ in Proposition so that ϕ0) e iθ ϕe iθ 0) e iθ ϕ0) is nonnegative. Letting g f and ψ ϕ satisfies the conclusion of the Corollary. As in Proposition.3 of [], if f and ϕ determine a bounded weighted composition operator on H 2 β) and if g and h are functions in H 2 β) such that gh is also in H 2 β), then W f,ϕ )gh) h ϕw f,ϕ g g ϕw f,ϕ h. Following the book of Cowen and MacCluer [9, p. 6], the generating function for the space H 2 β) is the function kz) j0 z j βj) 2 The function kz) is analytic on the unit disk and, for each w in the disk, the function K w z) kwz) belongs to H 2 β). The K w are the kernel functions for H 2 β), that is, the functions for which fw) f, K w and this implies K w 2 k w 2 ). First, we wish to characterize the functions f and ϕ of bounded Hermitian weighted composition operators. Theorem 3. Let k be the generating function for H 2 β). If W f,ϕ is a bounded Hermitian weighted composition operator on H 2 β), then f0) and ϕ 0) are real, fz) cka 0 z) ck a0 z) and ϕz) a 0 + a β) 2 z k a 0 z) ka 0 z) where a 0 ϕ0), a ϕ 0), and c f0). Proof. Let W f,ϕ be a bounded Hermitian weighted composition operator on H 2 β). Let w and z be points of the open unit disk, D. Now, from the definition of W f,ϕ, W f,ϕ K w )z) fz)k w ϕz)) fz)kwϕz)) and, using the special properties of W f,ϕ acting on kernel functions, W f,ϕ K w)z) fw)k ϕw) z) fw)kϕw)z) Because W f,ϕ is self-adjoint, we have ) fz)kwϕz)) fw)kϕw)z) Putting w 0, this means fz)k0ϕz)) f0)kϕ0)z) Recalling that we have assumed that k0) /β0) 2 and that a 0 ϕ0), we have fz) f0)ka 0 z)

5 Putting z 0 also, we get so c f0) is real, and we have HERMITIAN WEIGHTED COMPOSITION OPERATORS 5 f0) f0)ka 0 0) f0) fz) cka 0 z) ck a0 z) Since we are not interested in the case f 0 which gives W f,ϕ 0, we will assume c is a non-zero real number. Using this expression for f in Equation ), we have cka 0 z)kwϕz)) cka 0 w)kϕw)z) cka 0 w)kϕw)z) The Taylor coefficients of k are real numbers, so ku) ku) for any complex number u. The above equality can be rewritten as ka 0 z)kwϕz)) ka 0 w)kϕw)z) Taking the derivative with respect to z gives k a 0 z)a 0 kwϕz)) + ka 0 z)k wϕz))wϕ z) ka 0 w)k ϕw)z)ϕw) Letting z 0, we have k 0)a 0 kwϕ0)) + k0)k wϕ0))wϕ 0) ka 0 w)k 0)ϕw) or, noting that k 0) /β) 2 and ϕ 0) a, taking conjugates and solving for ϕ, we have ϕw) a 0 + a β) 2 w k a 0 w) ka 0 w) Finally, taking the derivative again, and setting w 0, we get ϕ 0) a β) 2 k a 0 0) ka 0 0) β)2 a k 0) k0) a Since ϕ 0) a, this latter equality says ϕ 0) ϕ 0) which means that ϕ 0) is real and the proof is complete. We prove the converse of the above theorem for weighted Hardy spaces where the point evaluation kernel is given by K w z) wz) κ for κ, that is, the generating function is kz) z) κ. For κ >, since the weighted Bergman space A 2 κ 2 consists of the same functions as H2 β κ ) and fz) 2 κ ) z 2 ) κ 2 da π D gives the norm for the space, it is clear that multiplication by an H function gives rise to a bounded operator on H 2 β κ ). It also follows that if f is in H 2 β κ ) and g is analytic on D with gz) fz) for all z in D, then g is also in H 2 β κ ). These facts and similar ones will be used below without special reference. Corollary 4. For κ, let H 2 β κ ) be the weighted Hardy space with kernel function K w z) wz) κ. If W f,ϕ is a bounded Hermitian weighted composition operator on H 2 β κ ), then fz) c a 0 z) κ ck ϕ0) z) and ϕz) a 0 + a z a 0 z

6 6 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO where a 0 ϕ0) is a complex number and a ϕ 0) is real number such that ϕ maps the unit disk into itself and c f0) is a non-zero real number. Proof. The generating function for H 2 β κ ) is kz) z) κ, so k0), as was assumed for the scaling. We also see that k z) κ z) κ so /β) 2 k 0) κ and β) 2 /κ. From this, we conclude that ϕz) a 0 + a β) 2 z k a 0 z) ka 0 z) a 0 + a κ z κ a 0z) κ a 0 z) κ a 0 + a z a 0 z Not surprisingly, this duplicates the corresponding implication of Theorem 2. of [] for the standard Hardy space for which κ. Since the function ϕ is independent of κ, Corollary 2.3 of [] describes the conditions on a 0 and a that result in the map ϕ taking the unit disk into itself. This result is recorded below. Lemma 5. Let a be real. Then ϕz) a 0 + a z/ a 0 z) maps the open unit disk into itself if and only if 2) a 0 < and + a 0 2 a a 0 ) 2 We are now ready to state the necessary and sufficient conditions for the functions f and ϕ that guarantee W f,ϕ is a bounded Hermitian weighted composition operator on H 2 β κ ). Theorem 6. For κ, let H 2 β κ ) be the weighted Hardy space with kernel function K w z) wz) κ. If the weighted composition operator W f,ϕ is bounded and Hermitian on H 2 β κ ), then f0) and ϕ 0) are real and ϕz) a 0 + a z/ a 0 z) and fz) c a 0 z) κ ck ϕ0) z) where a 0 ϕ0), a ϕ 0), and c f0). Conversely, suppose a 0 is in D, and suppose c and a are real numbers. If ϕz) a 0 + a z/ a 0 z) maps the unit disk into itself and fz) c a 0 z) κ, then the weighted composition operator W f,ϕ is Hermitian and bounded on H 2 β κ ). Proof. The first statement follows from Corollary 4. If a 0, a, c, f, and ϕ are as in the hypothesis, the results of [9, p. 27 and Section 3.] show that W f,ϕ is a bounded operator on H 2 β κ ). Also, a straightforward calculation shows that Equation ) holds for all z and w in the disk, which means that W f,ϕ is Hermitian. When a 0 0, the resulting operator is a constant multiple of a composition operator. In this case, W f,ϕ self adjoint implies ϕz) rz with r. For r ±, the composition operators are C ϕ ±I and for < r <, the composition operators are easily understood compact operators, so we will only consider non-zero values for a 0. Furthermore, without loss of generality, we may take a 0 to be a positive number by using the transformation in Corollary 2. With these hypotheses added, in the remainder of the paper, we will look at the different cases suggested by Lemma 5: + a 2 0 < a < a 0 ) 2, a + a 2 0, and a a 0 ) 2.

7 HERMITIAN WEIGHTED COMPOSITION OPERATORS 7 2. Compact and isometric Hermitian weighted composition operators We will see that the Hermitian weighted composition operators are all compact when the parameters defining ϕ satisfy + a 0 2 < a < a 0 ) 2 and they are isometries when a + a 0 2. Using Proposition, without loss of generality, we may assume 0 < a 0 <. Easy calculations see, for example, the discussion following Corollary 2.3 in []), show that when 0 < a 0 < and + a 2 0 < a < a 0 ) 2 the function ϕ maps the closed unit disk into the open unit disk. In particular, since ϕ is continuous and maps the interval [, ] into, ), it must have a fixed point in, ). Similarly, when a + a 2 0, we see that ϕz) a 0 + a z a 0 z a a2 0 )z a 0 z a 0 z a 0 z so that ϕ is an automorphism of the unit disk that satisfies ϕ ϕz) z. Moreover, since ϕ) and ϕ maps [, ] onto itself, it must have a fixed point in, ) in this case as well. The most important part of the argument in these cases depends on the presence of the fixed point. First, we show that in the first case, the operators are Hilbert-Schmidt and therefore they are compact operators. Theorem 7. For κ, let H 2 β κ ) be the weighted Hardy space with kernel function K w z) wz) κ. Suppose ϕz) a 0 + a z/ a 0 z) and fz) c a 0 z) κ ck ϕ0) z) where c is a real number. If 0 < a 0 < and + a 2 0 < a < a 0 ) 2 then W f,ϕ is Hilbert-Schmidt. Proof. Let f n z) βn) z n. Then {f n } n0 is an orthonormal basis of the weighted Hardy space H 2 β κ ). We will show that W f,ϕ f n ) 2 is finite which will show W f,ϕ is Hilbert-Schmidt [9, Theorem 3.23]. The comments above imply that in this case, there is r < so that ϕz) < r for z in D. Now, because the spaces H 2 β κ ) and A 2 κ 2 D) are equivalent, there are constants M and M so that W f,ϕ f j ) 2 M c 2 K a0 z) 2 βj) 2 ϕz) 2j z ) κ 2 da j0 j0 D π c 2 M a 0 ) 2κ βj) 2 r 2j z ) κ 2 da D π Thus c 2 a 0 ) 2κ M j0 n0 βj) 2 r 2j j0 c 2 a 0 ) 2κ M kr 2 ) W f,ϕ f n ) 2 is finite. n0

8 8 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO We will next explore the eigenvectors of these operators. Lemma 8. Suppose a 0 and a 0 are real numbers such that 0 < a 0 < and ϕz) a 0 + a z maps the disk into itself with fixed point α in the disk. a 0 z This implies 3) 4) 5) and αϕz) a 0α) αz) a 0 z ϕ α) a a 2 0 ) + a 0α a 0 α for gz) α z αz then Proof. That α is a fixed point of ϕ means α a 0 + 6) α a 0 α) α a 0 α 2 a α + a 0 a 2 0α and, rearranging the terms in this equation, we get 7) gϕz)) ϕ α)gz) a α a 0 α a α + a 0 a 2 0 α a 0 α α a 0 a α + a 0 α 2 a 2 0α αa + a 0 α a 2 0) If a + a 0 α a then α a 0 and a 0, contrary to our hypothesis, so we may also assume a + a 0 α a Similarly, if α were 0, we would have a 0 0, so α 0. The definition of ϕ means ϕz) a 0 a z/ a 0 z), so for z α, we get α a 0 ϕα) a 0 a α a 0 α This, together with Equation 7), gives a a 0 α α a 0 a + a 0 α a 2 0 α Taking the derivative of ϕ, we get ϕ α) a / a 0 α) 2 and we see ) ϕ a α) a 0 α) 2 a a 0 α a 0 α a + a 0 α a 2 0 a 0 α as we wanted to prove. Writing ϕ as ϕz) a 0 + a a 2 0 )z) / a 0 z), we get a0 + a a 2 0 αϕz) α )z ) a 0 z Using Equation 6), this becomes 8) a 0α) a α + a 0 a 2 0 α)z a 0 z αϕz) a 0α) α a 0 α)z a 0 z as we wished to show. a 0α) αz) a 0 z so

9 HERMITIAN WEIGHTED COMPOSITION OPERATORS 9 From the definition of g and Equation 8), we see gϕz)) α ϕz) αϕz) α ϕz)) a 0z) a 0 α) αz) α a 0 αa 0 z + a 2 0 z a z a 0 α) αz) α a 0 a z a 0 z ) a 0z) a 0 α) αz) α a 0) za + a 0 α a 2 0 ) a 0 α) αz) Using Equation 7) and our calculation of ϕ α) in this result, we get gϕz)) αa + a 0 α a 2 0 ) za + a 0 α a 2 0 ) a 0 α) αz) a + a 0 α a 2 ) ) 0 α z ϕ α)gz) a 0 α αz which completes the proof of the lemma. We are now ready to collect the information we have to characterize the eigenvectors and eigenvalues of these weighted composition operators. Theorem 9. For κ, let H 2 β κ ) be the weighted Hardy space with kernel function K w z) wz) κ. Suppose a 0 and a are real numbers such that 0 < a 0 < and ϕz) a 0 + a z maps the disk into itself with fixed point α in a 0 z the disk and fz) c a 0 z) κ ck ϕ0) z) for some real number c. For each non-negative integer j, the function g j z) αz) κ ) α z j αz is an eigenvector of the operator W f,ϕ with eigenvalue fα)ϕ α) j. Proof. Since g j is a bounded analytic map on the unit disk it belongs to H 2 β κ ). ) c α ϕz) j W f,ϕ g j )z) a 0 z) κ αϕz)) κ αϕz) Now using Lemma 8, we get, W f,ϕ g j )z) c a 0 z) κ a 0 z) κ a 0 α) κ αz) κ c ϕ a 0 α) κ α) ) j αz) κ z α ϕ α) αz α z αz We can apply this to the case in which W f,ϕ is compact. )) j ) j fα)ϕ α) j g j z) Corollary 0. For κ, let H 2 β κ ) be the weighted Hardy space with kernel function K w z) wz) κ. Suppose a 0, a, and c are real numbers such that 0 < a 0 < and + a 2 0 < a < a 0 ) 2. If ϕ, f, and g j are defined as in Theorem 9 and α is the fixed point of ϕ in the disk, then {g j } j0 is an orthogonal basis for H 2 β κ ) consisting of eigenvectors for W f,ϕ with W f,ϕ g j ) fα)ϕ α) j g j. In particular, W f,ϕ is a compact Hermitian operator and the spectrum of W f,ϕ is {0} {fα)ϕ α) j : j 0,, 2, }.

10 0 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO Not surprisingly, this agrees with Theorem of [6] which describes the spectra of compact weighted composition operators on H 2 β κ ). Proof. Since α is the fixed point of a non-automorphism of the unit disk, ϕ α) <, and each eigenvector given by Theorem 9 corresponds to a different eigenvalue of this self-adjoint operator. Thus {g j } is an orthogonal sequence of eigenvectors. Because the functions in H 2 β κ ) are analytic in a neighborhood of α, an easy power series argument shows the span of this sequence is dense in H 2 β κ ). We next look at the case + a 2 0 a, for which ϕz) a 0 + a z a 0 z a a2 0 )z a 0 z a 0 z a 0 z is an automorphism of the unit disk with ϕ ϕz) z. From this, we see that W 2 f,ϕ W f ϕ,ϕ ϕ T f ϕ, so it is worth looking at W 2 f,ϕ. Lemma. Suppose ϕz) a 0 + a z a 0 z and fz) c a 0 z) κ ck ϕ0) z) If 0 < a 0 <, a + a 2 0, and c a2 0 )κ/2 then W f,ϕ is an invertible isometry. Proof. Let g be in H 2 β κ ). Then, W 2 f,ϕ g)z) W f,ϕw f,ϕ g)z) W f,ϕ f g ϕ)z) f f ϕ gϕ ϕ)) z) fz) fϕz)) gϕϕz))) c c a 0 z) κ a a 0 z gz) c2 0 a 0 z )κ gz) a 2 gz) 0 )κ where the last equality holds by the choice of c in the hypothesis. The operator W f,ϕ is self-adjoint therefore Wf,ϕ 2 W f,ϕ W f,ϕ W f,ϕ W f,ϕ. Thus the above computation tells us that W f,ϕ W f,ϕ W f,ϕ W f,ϕ is the identity operator which means W f,ϕ is an invertible isometry. In this case, the fixed points of ϕ are ± a 2 0 )/a 0 and it is clear that α a 2 0 )/a 0 is inside the open disk. Hence ϕ is an automorphism of the unit disk with a fixed point inside the open unit disk. In the previous section, note that the hypotheses of Lemma 8 and of Theorem 9 refer to ϕ having an interior fixed point and therefore apply to the case we are considering. In complete analogy to Corollary 0, we characterize the Hermitian isometric weighted composition operators on H 2 β κ ). Corollary 2. For κ, let H 2 β κ ) be the weighted Hardy space with kernel function K w z) wz) κ. Suppose a 0, a, and c are real numbers such that 0 < a 0 <, a + a 2 0, and c a2 0 )κ/2. If ϕ, f, and g j are defined as in Theorem 9 and α is the fixed point of ϕ in the disk, then {g j } j0 is an orthogonal basis for H 2 β κ ) consisting of eigenvectors for W f,ϕ with W f,ϕ g j ) ) j g j. In

11 HERMITIAN WEIGHTED COMPOSITION OPERATORS particular, W f,ϕ is Hermitian and unitary and the spectrum of W f,ϕ is the set {, }. Proof. The map here is the map ϕ of Lemma 8 with a +a 2 0. From Lemma 8, a a 2 0 we get gϕz)) + a ) 0α α z a 0 α αz. Because a a 2 0, it follows that gϕz)) ) α z αz gz). If j is a non-negative integer g j z) αz) κ gz))j where gz) α z αz. Then, W f,ϕ g j )z) a2 0 )κ/2 a 0 z) κ αϕz)) κ gϕz)))j using Lemma 8 we get, W f,ϕ g j )z) a2 0 )κ/2 a 0 z) κ but α a 2 0 a 0 ) a 0 z κ ) α z ) j a 0 α) αz) αz hence a 0 α a 2 0. Thus W f,ϕ g j )z) ) j αz) κ ) α z j ) j g j z) αz In Corollary 0, we showed that {g j } j0 is an orthogonal basis for H2 β κ ), and for this reason, we know that the spectrum has no other elements besides ±. 3. Absolutely continuous Hermitian weighted composition operators To study the remaining cases of Hermitian weighted composition operators, when a a 0 ) 2 for 0 < a 0 <, we first show that the operators W f,ϕ belong to a continuous semigroup of Hermitian operators. Recall that an indexed collection {A t : t 0} of bounded operators is called a continuous semigroup of operators if A s+t A s A t for all non-negative real numbers s and t, A 0 I, and the map t A t is strongly continuous. Similarly, for 0 < θ π 2, an indexed collection {A t : arg t < θ} of bounded operators is called an analytic semigroup of operators if A s+t A s A t for arg s < θ and arg t < θ, the map t A t is analytic in the angular domain {t : t 0 and arg t < θ}, and lim t 0, arg t <θ A t h h, in norm, for all h. An easy calculation, presented in [] and other sources, shows the collection of weighted composition operators {T ft C ϕt } satisfies the semigroup property if and only if 9) f s f t ϕ s ) f s+t and 0) ϕ t ϕ s ϕ s+t

12 2 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO In other words, the composition operator factors in a semigroup of weighted composition operators form a semigroup of composition operators and the Toeplitz operator factors form a cocycle of Toeplitz operators. It is worthy of note that for each t with Re t 0, the linear fractional map ϕ t maps the unit disk into itself, properly if Re t > 0 and ϕ t is an automorphism if t is on the imaginary axis and for each such t, ϕ t ) ϕ t). Let P {t : Re t > 0} denote the right half plane. For t in P, let A t T ft C ϕt where ) f t z) + t tz) κ and t + t)z 2) ϕ t z) + t tz Note that the relationship between a 0 and t can be expressed as a 0 ϕ t 0) t/ + t) for a 0 2 < 2. However, a ϕ t0) + t) 2 is not real unless t is real, so A t is not Hermitian on H 2 β κ ) for t not real. It is somewhat tedious, but not difficult, to use the semigroup structure proved in Theorem 3 and the calculation A t A t to show that the A t are all normal operators, but we put this off to do it more easily later Corollary 9). Theorem 3. The A t, for t in the right half plane P, form an analytic semigroup of weighted composition operators on H 2 β κ ). Let be the infinitesimal generator of the semigroup {A t }. The domain of is D A {f H 2 β κ ) : z ) 2 f H 2 β κ )}. For such f, f)z) z ) 2 f z) + κz )fz) Proof. To show A t A s A t+s, it suffices to show that the cocycle relationship Equation 9)) and the semigroup relationship Equation 0)) hold. For the f t and ϕ t given above, the required equalities are f s z) f t ϕ s z)) + s sz) κ + t t s+ s)z +s sz )κ + s + t) s + t)z) κ f s+tz) and ϕ s ϕ t z)) s + s) t+ t)z +t tz + s s t+ t)z +t tz s + t) + s + t))z + s + t) s + t)z ϕ s+tz) Thus, the set {A t : Re t > 0} is a semigroup of weighted composition operators. Since operator valued functions are analytic in the norm topology if and only if they are analytic in the weak-operator topology Theorem 3.0. of [8, p. 93]), it is sufficient to check that the map t A t h, K z is analytic for each t in the right half plane. This is easy to see because h is analytic and A t h, K z f t z)hϕ t z)) which is clearly analytic in t for fixed z. + t tz) κ h t + t)z + t tz )

13 HERMITIAN WEIGHTED COMPOSITION OPERATORS 3 Finally, we must show the strong continuity at t 0. t /3, when z <, f t z) + t tz κ t z ) κ 2/3) κ 3κ First note that for so for t /3, we have f t 3 κ. Similarly, the results of Exercises 2..5 and 3..3 of [9] imply that the norms C ϕt are uniformly bounded if t /3 and t is in P, so, we see that the norms A t are uniformly bounded on this set. Observe that for α in the disk, A t K α )z) f t z)k α ϕ t z)) + t tz αt + t)z)) κ As t approaches 0, the functions A t K α converge uniformly, and therefore in H 2 β κ ), to K α. Since the kernel functions have dense span in H 2 β κ ) and the norms A t are uniformly bounded for t in P with t /3, it follows that for each f in H 2 β κ ), we also have lim t 0,t P A t f f. Thus, A t is strongly continuous at t 0 and the proof that {A t } is an analytic semigroup is complete. To consider the infinitesimal generator, let f be a function in H 2 β κ ) and suppose z is a point of the disk. Then we have f)z) lim t 0 + t A f t z)fϕ t z)) fz) t I)f) z) lim t 0 + t lim f tz) fϕ tz)) fz) t 0 + ϕ t z) z ϕ t z) z t + f tz) fz) t f z) z) 2 κ z)fz) z ) 2 f z) + κz )f This means that if f is in the domain D A of, then z ) 2 f z) + κz )fz) must be in H 2 β κ ). Since the operator of multiplication by z is bounded on H 2 β κ ), when f is in H 2 β κ ), z )f is also in H 2 β κ ), so we conclude that if f is in the domain of, then z ) 2 f f) z )f must be in H 2 β κ ). To complete the proof, we must show that if f is a function in H 2 β κ ) such that z ) 2 f is also in H 2 β κ ), then f is in the domain of. Clearly, every polynomial is in D A because for a polynomial, the convergence of the limit in the calculation above is uniform on the closed disk, and therefore is a limit in the H 2 β κ ) norm. The remainder of the proof closely follows the proof of the determination of the domain of the infinitesimal generator in []. We very briefly outline and update that proof here. Suppose f is a function in H 2 β κ ) and suppose z ) 2 f is also in H 2 β κ ). Letting fz) k0 a kz k, a straightfoward calculation gives z ) 2 f z) a + 2a + 2a 2 )z + k )a k 2ka k + k + )a k+ ) z k k2 Since z ) 2 f is also in H 2 β κ ), this series converges in H 2 β κ ) norm to z ) 2 f.

14 4 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO For each positive integer n, let p n be the polynomial n n k p n z) n a kz k k0 which is the polynomial approximation to f suggested by Cesáro summation. Then lim n p n f and the convergence is in the H 2 β κ ) norm. In addition, we let r n be the polynomial r n z) z ) 2 p nz). Now, let q n be the polynomial q n z) a + n n 2a + 2a 2 )z + n k2 n k n k )a k 2ka k + k + )a k+ ) z k which is the polynomial approximation to z ) 2 f suggested by Cesáro summation. Then lim n q n z ) 2 f and the convergence is in the H 2 β κ ) norm. In [], it was proved that lim n q n r n H 2 0. The norm of any polynomial p in H 2 β κ ) satisfies p H 2 β κ) p H 2, so we have lim n q n r n H 2 β κ) 0. Since lim n q n z ) 2 f, this means that lim n r n z ) 2 f also. Now, lim n p n f which implies, because multiplication by z is bounded, that lim n z )p n z )f. We note that p n ) z ) 2 p n +κz )p n r n + κz )p n which means that lim n p n ) lim n r n + lim n κz )p n z ) 2 f + κz )f. Thus, we have lim n p n f and lim n p n ) z ) 2 f + κz )f. Since is a closed operator, this means that f is in D A, the domain of, and that f) z ) 2 f + κz )f. We want to find the potential eigenvectors and eigenvalues of the infinitesimal generator and consider their contribution to the spectra of and the A t. Lemma 4. If f is analytic in the disk and z ) 2 f + κz )f λfz), then for some constant C. fz) f λ z) Proof. If z ) 2 f + κz )f λfz), then C z) κ e λ z z ) 2 f z) κz ) + λ) fz) Thus f z) fz) κ z + λ z ) 2 Integrating both sides, we get ln fz) κ lnz ) λ z + c for some constant c. This implies fz) C z) κ e λ z

15 for some constant C. HERMITIAN WEIGHTED COMPOSITION OPERATORS 5 Corollary 5. For κ, the functions f λ of Lemma 4 do not belong to H 2 β κ ), and for t > 0, the operator A t has no eigenvalues. Proof. Because {A t } t 0 is a strongly continuous semigroup of bounded operators, σ p A t ) e tσp ) {0} and specifically, if λ is in σ p ), then e λt is in σ p A t ). See [22, p. 46], for example.) Because A t T ft C ϕt and both the Toeplitz operator and the composition operator are one-to-one, 0 is not in the point spectrum of A t. From Lemma 4, we know that λ is in σ p ) if and only if f λ z) z) κ e λ z H 2 β κ ) and if f λ z) is in H 2 β κ ) then the above containment means that e λt is an eigenvalue of A t. Now for t > 0, the operators A t are Hermitian and this means any eigenvalues must be real and e λt real for all t > 0 implies λ is real. In this case, if ν < λ, then f ν z) f λ z) for all z in D, so f ν is also in H 2 β κ ) and e νt is also an eigenvalue of A t. For self-adjoint operators on a Hilbert space, the eigenvectors corresponding to distinct eigenvalues are orthogonal. Since the number of real numbers less than λ is uncountable and H 2 β κ ) is a separable Hilbert space, it is not possible for e νt to be an eigenvalue for every ν < λ, so it is not possible for e λt to be an eigenvalue for any real number λ. It follows that no f λ is in H 2 β κ ), and that the operators A t have no eigenvalues. Besides being part of an analytic semigroup, these Hermitian weighted composition operators have another special property: each of these operators is cyclic on H 2 β κ ). Theorem 6. For κ, let H 2 β κ ) be the weighted Hardy space with kernel functions K w z) wz) κ. If W f,ϕ is the Hermitian weighted composition operator on H 2 β κ ) with fz) a 0 z) κ and ϕz) a 0 + a z a 0 z where 0 < a 0 < and a a 0 ) 2, then W f,ϕ is a star) cyclic Hermitian operator, indeed, the vector in H 2 β κ ) is a star) cyclic vector for W f,ϕ. Proof. We note that because W f,ϕ is Hermitian, a vector is a star-cyclic vector exactly when it is a cyclic vector. Since 0 < a 0 < and a a 0 ) 2, for t a 0 / a 0 ) we have W f,ϕ A t in the notation above. For w in the unit disk, A t K w C ϕ t T f t K w f t w)k ϕtw) f t w)k ϕtw) Since the vector in H 2 β κ ) is K 0, we have A t ) f t 0)K ϕt0). Now, A t ft 0)K ϕt0)) At A t )) A 2t ) f 2t 0)K ϕ2t 0) and in general, clearly, A n t ) f nt 0)K ϕnt0). To check cyclicity, we need to investigate the span of these vectors. Since the factor f nt 0) is just a non-zero number, is a cyclic vector for A t if and only if

16 6 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO span{k ϕnt0)} is dense in H 2 β κ ). This span is dense if and only if 0 is the only vector orthogonal to all the vectors K ϕnt0). Since h, K ϕnt0) hϕ nt 0)), this means that the span is dense if and only if the only function h in H 2 β κ ) such that hϕ nt 0)) 0 for n, 2, 3, is the zero function. Note that since t is real and positive, each of the points ϕ nt 0) nt/ + nt) is in the interval [0, ) on the real axis. Because K w z) wz) κ, we see K w w 2 ) κ/2 and by the Cauchy- Schwartz inequality, for each h in H 2 β κ ), there is a constant C so that for all w in the unit disk, hw) h, K w h K w h w 2 ) κ/2 C exp{ w ).2 } This means that every function in H 2 β κ ) satisfies the growth condition in the hypothesis of a theorem of Shapiro and Shields [25] or see [2, p. 6]) on zero sets of analytic functions on the disk. It follows from this theorem that the set of points {ϕ nt 0)} is the zero set of a non-zero function in H 2 β κ ) if and only if the sequence is a Blaschke sequence. To check this, consider the sum ϕ nt 0) ) n n nt + nt ) n + nt Since this sum is infinite, the sequence is not a Blaschke sequence and therefore the vectors {K ϕnt0)} have dense span in H 2 β κ ). This means is a cyclic vector for A t W f,ϕ. We will see that the estimate used in the proof of Theorem 3 above for A t can be improved: in fact, it follows from Theorem 7 and Corollary 9 that A t for all t in P. Theorem 6 says that, for t > 0 i.e. 0 < a 0 ), each of the operators A t is cyclic, which means that each is unitarily equivalent to an ordinary multiplication operator see, for example, [2, p. 269]). The following result gives this explicitly. The results here are similar to those in the Hardy space case as presented in [], but the strategy here will be to use the weighted composition operators to discover facts about the invariant subspaces in the weighted Bergman spaces, rather than using facts about invariant subspaces of the Hardy space to get properties of the operators A t. The relevant L 2 space is L 2 [0, ], µ) where µ is the absolutely continuous probability measure [4, p. 58] given by dµ ln/x))κ Γκ) In the theorem below, the multiplication operator M h is defined as usual: M h g) x) hx)gx) for g in L 2 [0, ], µ). Theorem 7. For each t in P, the multiplication operator M x t on L 2 [0, ], µ) and the weighted composition operator A t on H 2 β κ ) are unitarily equivalent. In fact, the operator U : H 2 β κ ) L 2 given by, for s in P, is unitary and satisfies UA t M x tu. Uf s ) x s dx

17 HERMITIAN WEIGHTED COMPOSITION OPERATORS 7 Proof. First note that functions in L [0, ]) are in L 2 [0, ], µ) so the Lebesgue Dominated Convergence Theorem shows that uniform convergence on [0, ] implies norm convergence in L 2 [0, ], µ). It also follows that if h is in L [0, ]) then M h is a bounded operator on this Hilbert space with norm M h h. The Stone-Weierstrass Approximation Theorem shows that {x n } n0,,2,, has dense span in L 2 [0, ], µ). In particular, this means the span of {x s : s P} is dense in L 2 [0, ], µ). In the proof of Theorem 6, we saw that f n z) f n 0)K ϕn0)z) for n 0,, 2,, has dense span in H 2 β κ ), so the span of {f s : s P} is dense in H 2 β κ ). Thus, if U satisfies Uf s ) x s, we see that for each s in P, using Equation 9), we have UA t f s ) Uf s+t x s+t M x tx s M x tuf s ) Since the span of {f s : s P} is dense in H 2 β κ ), we conclude UA t M x tu. We will show that U is isometric on the span of {f s } for s in P. For r and s in P, we have f r, f s f r z), + s ) κ + s + s sz ) κ f r z), K s z) +s ) κ ) κ f r z), + s + r r s +s) + r) + s) r s) κ + s + r) κ. ) κ Similarly, for r and s in P, in L 2 [0, ], µ), we have x r, x s L 2 0 x r ln x)κ xs dx Γκ) Γκ) Γκ) Γκ) + s + r) κ + s + r) κ. 0 s +s z ) κ x r+ s ln x) κ dx When κ is a positive integer the integral above can be evaluated by applying integration by parts repeatedly but the general integral can be found in integral tables, for example, see [4, p. 58].) Now, the vectors {f s } have dense span in H 2 β κ ) and the vectors {x s } have dense span in L 2 [0, ], µ). The inner product calculations above show that U is isometric on these spans and therefore has a unique extension to an isometric

18 8 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO operator of H 2 β κ ) onto L 2 [0, ], µ), that is, it is a unitary operator between these spaces. Corollary 8. For each t with t > 0, the Hermitian weighted composition operator A t is unitarily equivalent to M x t. In particular, for each t > 0, these operators satisfy A t and have spectrum σa t ) [0, ]. Proof. For each t with 0 < t <, Theorem 7 implies A t U M x tu so and A t M x t sup{ x t : 0 x } σa t ) σm x t) {x t : 0 x } [0, ] Corollary 9. For each t in the right half plane P, the weighted composition operator A t is unitarily equivalent to M x t. In particular, for each t in the right half plane P, the weighted composition operator A t is normal and satisfies A t A t. Moreover, for t a + ib with a > 0, these operators satisfy A t and have spectrum that is a spiral, σa t ) {0} {e ay e iby : y 0}. Proof. For each t in the right half plane P, Theorem 7 implies A t U M x tu. Multiplication operators M h are normal for every h in L [0, ]) and satisfy M h M h. Since unitary equivalence preserves normality and adjoints, we have and A t U M x tu) U M x tu U M x tu A t A t M x t sup{ x t : 0 x } sup{x a : 0 x } and, letting x e y for 0 < x σa t ) σm x t) {x t : 0 x } {0} {e yt : y 0} {0} {e ay e iby : y 0} As we would expect, from the Hermitian and the non-hermitian, normal cases, t imaginary corresponds to A t being unitary. Such weighted composition operators have been considered by Forelli [3] who showed that they are isometries of H 2 and Bourdon and Narayan [] who showed that they are unitary on H 2. Corollary 20. For each t imaginary, the weighted composition operator A t is unitarily equivalent to M x t, which is a unitary operator. In particular, for each t on the imaginary axis, the weighted composition operator A t is unitary and satisfies A t A t A t. Moreover, for such t, these operators have spectrum the unit circle, σa t ) {e iy : y 0}. Proof. The calculations in the proof of Theorem 3 that show A s A t A s+t do not depend on Re t > 0, so the equation A t A t A t shows that A t A t A t A t I and A t is unitary for t on the imaginary axis. For each t in the right half plane P, Theorem 7 says UA t M x tu. Writing t a + ib and taking weak operator limits we see that UA ib w-lim a 0 + UA a+ib w-lim a 0 + M x a+ibu M x ibu

19 HERMITIAN WEIGHTED COMPOSITION OPERATORS 9 so A t and M x t are unitarily equivalent also for t on the imaginary axis. The spectral result is correct because σm x t) {x t : 0 < x }. In the case of the usual Hardy space, H 2, the spectral subspaces for the weighted composition operators were [] the subspaces χh 2 where χ is an atomic singular inner function with atom at {}. Our goal will be to prove the analogue of this result for the weighted Bergman spaces H 2 β κ ) A 2 κ 2. We define subspaces H c of H 2 β κ ) A 2 κ 2 as follows: Let H 0 H 2 β κ ). For c a negative real number, define the subspace H c by 3) H c closure {e c +z z f : f H 2 β κ )} These subspaces are well known and have been studied because they are relatively simple invariant subspaces for multiplication by z on H 2 β κ ) that are not associated with a zero set. The properties of these subspaces that we need are summarized in Lemma 2 below, but we will not prove it. Good sources for this kind of result are the books of Hedenmalm, Korenblum, and Zhu [7], especially Section 3.2, pp , or Duren and Schuster [2], Section 8.2, where proofs are given for the usual unweighted) Bergman space. Other sources are Theorem 2 of Roberts [23] or Korenblum [9]. Lemma 2. For each negative number c, the subspace H c is a proper subspace of H 2 β κ ) and if c and c 2 are negative real numbers with c 2 < c, then H c2 is a proper subspace of H c. Finally, c 0 H c 0). The idea underlying the focus on these subspaces is that the eigenvectors of our operators, if they had any eigenvectors, should be z) κ e λ e λ 2 z z) κ e λ +z 2 z If these were eigenvectors for A, say, then the spectral subspace associated with [0, r] for 0 r would be spanned by the eigenvectors whose eigenvalues are in [0, r]. This will be the case for A if and only if λ is a number so that 0 < e λ r. So suppose r is given and λ 0 ln r so that e λ 0 r. Looking at the eigenvectors, it looks like the subspace containing the eigenvectors for the eigenvalues with 0 < e λ < r ought, if they were actually in H 2 β κ ), to span the subspace e λ 0 +z 2 z H 2 β κ ) H ln r)/2. Theorem 22 below shows directly that these are invariant subspaces for the A t. We want to prove that these are, indeed, the spectral subspaces we are looking for. In [], this was accomplished for the Hardy space by using the special properties of these well-studied subspaces to identify the projections and construct the spectral integrals. However, in the context of H 2 β κ ) A 2 κ 2, these subspaces are not as well understood as in the Hardy space. Instead, we will reverse the strategy: we will use the unitary U to carry information from the multiplication operators on L 2 [0, ], µ), which are well understood, back to the weighted composition operators on H 2 β κ ), and use this information to better understand these invariant subspaces of the Bergman spaces. Theorem 22. For 0 t and c 0, the subspace H c is invariant for A t.

20 20 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO Proof. Suppose f is in H 2 β κ ) so that expc + z)/ z))) f is in H c. Then t+ t)z + ) +t tz A t e c +z c z f + t tz) κ e t+ t)z +t tz f ϕ t +z c2t+ + t tz) κ e z ) ) f ϕ t e c +z e 2ct z + t tz) κ f ϕ t e c +z z At e 2ct f) and we see this latter vector is also in H c. Since {expc + z)/ z))) f : f H 2 β κ )} is dense in H c and H c is closed, this shows that H c is invariant for A t. For t > 0, the functions x t are increasing for 0 x. This means the spectral subspaces for the Hermitian) multiplication operators M x t are just the subspaces L δ {f L 2 [0, ], µ) : fx) 0 for δ < x }, for 0 δ. Obviously, L 0 0), L L 2, for δ < δ 2, the subspaces satisfy L δ L δ2, and the orthogonal projection of L 2 onto L δ is the multiplication operator M χ[0,δ]. We want to identify U L δ in H 2 β κ ). Theorem 23. For c < 0 and 0 < δ <, let H c and L δ be as above. If U is the unitary operator of H 2 β κ ) onto L 2 [0, ], µ) as in Theorem 7, then U L δ H ln δ)/2 or equivalently UH c L e 2c Proof. Clearly the two versions of the conclusion are equivalent; we will prove the first. Since this result follows directly from the work of [] for H 2 when κ, we may assume that κ >. For 0 < a <, let f a be the function defined by { ln/x)) κ+ 0 < x a f a x) 0 a < x It is not difficult to see that lim x 0 + f a x) 0, that f a is continuous and increasing on 0, a] and that therefore f a is in L [0, ]) and also in L 2 [0, ], µ). For t > 0, computing in L 2, we have f a, x t Γκ) a 0 ln/x)) κ xt dµx) Γκ) Let F a U f a, and computing in H 2 β κ ), we have a t+ Γκ) + t a f a, x t U f a, U x t F a, 0 a t+ x t dx Γκ) t + + t tz) κ + t) κ F a, t +t z)κ + t) κ F t a + t ) where the last equality holds because t is positive exactly when t/ + t) is a point of 0, ) in the unit disk and the inner product is the point evaluation of F a.

21 HERMITIAN WEIGHTED COMPOSITION OPERATORS 2 Rewriting the equality, we have, for every t > 0, t F a + t ) Γκ) + t)κ a t+ Since F a is analytic in the disk, the equality for all t > 0 gives equality whenever z is in the disk. In particular, we find t +t F a z) Γκ) z) κ e ln a z e ln a)/2 Γκ) z) +z a)/2) eln z ) κ If sx) is a simple function defined on [0, ] such that s {y}) is an open interval for each y in the range of s, we can easily approximate s in L with linear combinations of functions of the form f a f b for 0 b < a. It follows that the closure in L 2 [0, ], µ) of the span of {f a : 0 < a δ} is L δ. Thus, we have U L δ ) span{u f a : 0 < a δ} { e ln a)/2 span Γκ) z) { span z) +z a)/2) eln κ +z a)/2) eln κ z ) : 0 < a δ } z ) : 0 < a δ H ln δ)/2 } For 0 < r, let P r be the orthogonal projection of H 2 β κ ) onto the subspace H ln r)/2 and let P 0 0. The set of projections {P r } 0 r form a resolution of the identity and each commutes with each A t because each A t is Hermitian and each H ln r)/2 is invariant for A t Theorem 22). Corollary 24. Let t be a positive real number. Letting P 0 0, the projections {P r } 0 r form a resolution of the identity for the operator A t. Related to A t, the projection P r corresponds to the interval [0, r t ] as a subset of the spectrum of A t. This means we have A t 0 r t dp r Proof. For each t with t > 0, the subspace L δ M χ[0,δ] L 2 is associated with {x : 0 x t δ}, so M x t 0 δt dm χ[0,δ]. The unitary U implements an equivalence between A t and M x t, so L δ is associated with H ln r)/2 and A t 0 rt dp r. 4. Extremal functions for invariant subspaces determined by atomic singular inner functions Suppose N is a subspace of the Bergman space A 2 κ 2 H2 β κ ) that is invariant for the operator of multiplication by z. If there are functions f in N so that f0) 0 and G is a function of N so that 4) G and G0) sup{re f0) : f N and f }

22 22 CARL C. COWEN, GAJATH GUNATILLAKE, AND EUNGIL KO then we say G solves the extremal problem for the invariant subspace N and we say G is an A 2 κ 2 inner function. For more information about such extremal problems, see [7, p. 56] or [2, p. 20 or pp ], for example.) The subspaces H c defined by Equation 3) for c < 0 are of interest to us because they are the spectral subspaces for the operators A t, but they are of wider interest because they are invariant for multiplication by z and they are associated with the atomic singular inner functions e c+z)/ z) for c < 0. In the study of invariant subspaces for the Bergman shifts, it is important to solve the extremal problem of Equation 4), but it is not always straightforward to do so. We show that the unitary equivalence of Theorem 7 can be used to solve these problems for the subspaces H c. Note first that fz) e c+z)/ z) is in H c and f0) e c 0, so the formulation of the extremal problem given above is applicable. Our computation of the extremal functions requires the use of the incomplete Gamma function [4, p. 949] usually defined as 5) Γa, w) w t a e t dt where a is a complex parameter and w is a real parameter. An alternate definition [4, p. 950] in which both a and w are complex parameters is 6) Γa, w) e w w a e wu + u) a du 0 In our situation, we have a 0 and for this case, we see that Γa, w) is analytic for Re w > 0. Theorem 25. For c < 0, if H c is the invariant subspace of the weighted Bergman space A 2 κ 2 H2 β κ ) defined by then the extremal function for H c is 7) G c z) H c closure{e c +z z f : f A 2 κ 2 H 2 β κ )} Γκ, 2c/ z)) Γκ) Γκ, 2c) Proof. The Cauchy-Schwarz inequality implies that to maximize Re g, f where f is given and g is a unit vector in the subspace N, then g ζp f/ P f where P is the orthogonal projection onto N and ζ is chosen with ζ so that g, f > 0. Suppose g is in H c. Then Re g0) Re g, K 0 Re g, f 0, where f 0 f s for s 0 as in Equation ). Let G c be the function in H c that solves this extremal problem, that is, G c ζp e 2cf 0 / P e 2cf 0 for ζ chosen with ζ so that G c 0) > 0. We want to find an explicit description of G c. We will ignore ζ for the moment, and at the end of the proof, we will find an appropriate choice for ζ. We want to move the problem to L 2 ; we have g, f 0 Ug, Uf 0 Ug, x 0 Ug, and g is in H c if and only if Ug is in L e 2c. Writing δ for e 2c, we see that UG c Q δ )/ Q δ ) where Q δ is the projection of H 2 β κ ) onto L δ, that is, Q δ M χ[0,δ]. It follows that UG c χ [0,δ] χ [0,δ]

23 HERMITIAN WEIGHTED COMPOSITION OPERATORS 23 Now, to find the necessary norm, 8) P e 2cf 0 2 χ [0,δ] 2 δ 0 0 ln/x)) κ Γκ, 2c) Γκ) Γκ) χ [0,δ] 2 dµ δ 0 dµ dx y κ ) e y dy Γκ) 2c where the penultimate equality comes from the change of variables x e y and the last equality is the definition of the incomplete Gamma function [4, p. 949]. Similarly, to find the function itself, for s > 0, we have 9) P e 2cf 0 G c, f s P e 2cf 0 UG c, Uf s χ [0,δ], x s δ 0 x s ln/x))κ Γκ) s + ) κ Γκ) 2cs+) δ 0 x s dµ dx y κ ) e s+)y dy Γκ) 2c u κ ) e u du Γκ, 2cs + )) s + ) κ Γκ) On the other hand, we can find the value of G c, f s another way because f s z) + s sz) κ + s) κ s/s + )z) κ + s) κ K s/s+) where K α is the kernel for evaluation at α. Using this, we see 20) G c, f s G c, + s) κ K s/s+) + s) κ G c, K s/s+) G cs/s + )) + s) κ Putting the results of Equations 8), 9), and 20) together, we get P e 2cf 0 G cs/s + )) Γκ, 2cs + )) + s) κ P e 2cf 0 G c, f s s + ) κ Γκ) and solving for G c, we have ) s G c s + Γκ, 2cs + )) Γκ) Γκ, 2c) Examining this result, we have assumed s > 0, so s/s + ) is a number in 0, ) and c < 0, so 2cs + ) is a number on the positive real axis. If z satisfies 0 < z < and z s/s + ), then 2c + s) 2c/ z) and this result can be rewritten as Γκ, 2c/ z)) 2) G c z) Γκ) Γκ, 2c) The function G c is in H 2 β κ ), so it is analytic in the unit disk. The formulation of the incomplete Gamma function in Equation 6) is analytic in the right half plane, so if z is in the disk then 2c/ z) is in the right half plane. Thus, we see that G c z) and the expression on the right side of Equation 2) are both analytic for z in the disk and they agree when 0 < z <. This means that they are equal