Curvature inequalities for operators in the CowenDouglas class of a planar domain
Abstract.
Fix a bounded planar domain If an operator in the CowenDouglas class admits the compact set as a spectral set, then the curvature inequality where is the Sz̈ego kernel of the domain is evident. Except when is simply connected, the existence of an operator for which for all in is not known. However, one knows that if is a fixed but arbitrary point in then there exists a bundle shift of rank say depending on this such that We prove that these extremal operators are uniquely determined: If and are two operators in each of which is the adjoint of a rank bundle shift and for a fixed in then and are unitarily equivalent. A surprising consequence is that the adjoint of only some of the bundle shifts of rank occur as extremal operators in domains of connectivity These are described explicitly.
Key words and phrases:
CowenDouglas class, bundle shifts, curvature, Sz̈ego kernel, weighted kernel function2010 Mathematics Subject Classification:
47A, 47A25, 47B32, 30C, 30A311. Introduction
Let be a bounded, open and connected subset of the complex plane Assume that the boundary of consists of analytic Jordan curves. Let denote the boundary components of We shall always let denote the curve whose interior contains Set which is again a planar domain whose boundary consists of analytic Jordan curves. In this paper we study operators in first introduced by Cowen and Douglas in [5], namely, those bounded linear operators acting on a complex separable Hilbert space for which and which meet the following requirements.


and

These conditions ensure that one may choose an eigenvector with eigenvalue for any operator in such that is holomorphic on (cf. [5, Proposition 1.11]). This is the holomorphic frame for the operator Cowen and Douglas also provide a model for the operators in the class which is easy to describe:
If then is unitarily equivalent to the adjoint of the operator of multiplication by the coordinate function on some Hilbert space consisting of holomorphic function on possessing a reproducing kernel . Throughout this paper, we let denote the operator of multiplication by the coordinate function and as usual denotes its adjoint.
The kernel is complex valued function defined on which is holomorphic in the first and antiholomorphic in the second variable and is positive definite in the sense that is positive definite for every subset of the domain We will therefore assume, without loss of generality, that an operator in has been realized as the operator on some reproducing kernel Hilbert space The curvature of the operator is defined as
where and are the vectors
in Thus the curvature is a real analytic function on
It turns out that this definition of curvature is independent of the representation of the operator as the adjoint of a multiplication operator on some reproducing kernel Hilbert space Indeed, if also admits a representation as the adjoint of the multiplication operator on another reproducing kernel Hilbert space then we must have for some holomorphic function defined on (See [5, Section 1.15]). This implies
If and are any two operators in then any intertwining unitary must map the holomorphic frame of one to the other. From this, it follows that the curvatures of these two operators must be equal, as shown in [5, Theorem 1.17] along with the nontrivial converse.
Theorem 1.1.
Two operators and in are unitarily equivalent if and only if their associated curvature functions are equal that is for all
Recall that a compact subset is said to be a spectral set for an operator in if
where denotes the algebra of rational function whose poles are off and denotes the sup norm over the compact subset Equivalently, is a spectral set for the operator if the homomorphism defined by the formula is contractive. There are plenty of examples where the spectrum of an operator is a spectral set, for instance, this is the case for subnormal operators (See [8, Chapter 21]).
Now assume the closure of is a spectral set for the operator in The space is an invariant subspace for Representing the restriction of the operator to this subspace with respect to an orthonormal basis as a matrix, we have
It follows that is also a spectral set for For any in it is not hard to verify that
Since
where the Sz̈ego kernel of is the reproducing kernel for the Hardy space a curvature inequality becomes evident (see [11, Corollary 1.2]), that is,
(1.1) 
Equivalently, since the curvature inequality takes the form
(1.2) 
The operator on the Hardy space is in The closed unit disc is a spectral set for the operator The reproducing kernel of the Hardy space, as is wellknown, is the Sz̈ego kernel of the unit disc It is given by the formula for all in The computation of the curvature of the operator is straightforward and is given by the formula
Since the closed unit disc is a spectral set for any contraction, it follows that the curvature of the operator on the Hardy space dominates the curvature of every other contraction in
If the region is simply connected, then using the Riemann map and the transformation rule for the Sz̈ego kernel (See [3, Theorem 12.3]) together with the chain rule for composition, we see that
(1.3) 
This shows that in the case of bounded simply connected domain with jordan analytic boundary, the operator on is an extremal operator.
On the other hand, if the region is not simply connected, then (1.3) fails. Indeed, Suita (see [15]) has shown that
(1.4) 
Or equivalently,
(1.5) 
where is the adjoint of the multiplication by the coordinate function on the Hardy space We therefore conclude that if is not simply connected, then the operator fails to be extremal.
We don’t know if there exists an operator in admitting as a spectral set for which
The question of equality at just one fixed but arbitrary point in was answered in [11, Theorem 2.1]. An operator in which admits as a spectral set would be called extremal at if
Equivalently, representing the operator as the operator on a Hilbert space possessing a reproducing kernel we have that
(1.6) 
Since the the operator on the Hardy space is extremal, that is, for all and for every in the class of contractions in and the curvature is a complete invariant, one may ask:
Question 1.2 (R. G. Douglas).
If for some fixed in then does it follow that must be unitarily equivalent to ?
This question has an affirmative answer if, for instance, is a homogeneous operator and it is easy to construct examples where the answer is negative (cf. [10]).
The operator on the Hardy space is a pure subnormal operator with the property: the spectrum of the minimal normal extension, designated the normal spectrum, is contained in the boundary of the spectrum of the operator These properties determine the operator uniquely up to unitary equivalence. The question of characterizing all pure subnormal operator with spectrum and normal spectrum contained in the boundary of is more challenging if is not simply connected. The deep results of Abrahamse and Douglas (cf. [2, Theorem 11]) show that these are exactly the bundle shifts, what is more, they are in one to one correspondence with the equivalence classes of flat unitary bundles on the domain It follows that adjoint of a bundle shift of rank lies in Since bundle shifts are subnormal with spectrum equal to it follows that is a spectral set for the adjoint of the bundle shift. In fact, the extremal operator at found in [11], is the adjoint of a bundle shift of rank . Therefore, one may ask, following R. G. Douglas, if the curvature of an operator in admitting as a spectral set, equals then does it follow that is necessarily unitarily equivalent to the extremal operator at found in [11]. In this paper, we show that an extremal operator must be uniquely determined within where denotes the unitary equivalence class.
2. Preliminaries on bundle shifts of rank one
Let be an element in that is it is a homomorphism from the fundamental group of into the unit circle Such homomorphism is also called a character. Each of these character induce a flat unitary bundle of rank on (cf. [4, Proposition 2.5]). Following theorem establishes one to one correspondence between and the set equivalence classes of flat unitary vector bundle over of rank (See [7, p. 186].)
Theorem 2.1.
Two rank one flat unitary vector bundle and are equivalent as flat unitary vector bundle if and only if their inducing characters are equal that is
First, if is a holomorphic section of the bundle then for where is a coverng of we have that Thus the function is well defined on all of and is subharmonic there. Let be the linear space of those holomorphic sections of such that the subharmonic function on is majorized by a harmonic function on While there is no natural inner product on the space Abrahamse and Douglas define an inner product relative to the harmonic measure with respect to a fixed but arbitrary point We make the comment in [2, p. 118] explicit in what follows. Let be a regular exhaustion that is, it is a sequence of increasing subdomains of satisfying




boundary of each consists of finitely many smooth jordan curve.
Then norm of the section in is then defined by the limit
where denote the green function for the domain at the point and denote the directional derivative along the outward normal direction w.r.t the positively oriented boundary of The linear space is complete with respect to this norm making it into a Hilbert space. A bundle shift is simply the operator of multiplication by the coordinate function on
Theorem 2.2 (Abrahamse and Douglas).
Let and be rank one flat unitary vector bundles induced by the homomorphisms and respectively. Then the bundle shift is unitarily equivalent to the bundle shift if and only if and are equivalent as flat unitary vector bundles.
It is not very hard to verify that is a pure cyclic subnormal operator with spectrum and normal spectrum In fact these are the characterizing property for the rank one bundle shift.
Theorem 2.3 (Abrahamse and Douglas).
Every pure cyclic subnormal operator with spectrum and normal spectrum contained in is unitarily equivalent to a bundle shift for some character
Bundle shifts can also be realized as a multiplication operator on a certain subspace of the classical Hardy space Let be a holomorphic covering map satisfying Let denote the group of deck transformation associated to the map that is As is isomorphic to the fundamental group of , every character induce a unique element in By an abuse of notation we will also denote it by A holomorphic function on unit disc satisfying , is called a modulus automorphic function of index Now consider the following subspace of the Hardy space which consists of modulus automorphic function of index , namely
Let be the multiplication operator by the covering map on the subspace Abrahamse and Douglas have shown in [2, Theorem 5] that the operator is unitarily equivalent to the bundle shift
There is another way to realize the bundle shift as a multiplication operator on a Hilbert space of multivalued holomorphic function defined on with the property that its absolute value is single valued. A multivalued holomorphic function defined on with the property that its absolute value is single valued is called a multiplicative function. Every modulus automorphic function on induce a multiplicative function on namely, Converse is also true (See [17, Lemma 3.6]). We define the class consisting of multiplicative function in the following way:
So the linear space is consisting of those multiple valued function on for which is single valued, has a harmonic majorant on and is locally holomorphic in the sense that each point has a neighbourhood and a single valued holomorphic function on with the property on (See [6, p.101]). Since the covering map lifts the harmonic measure on at the point to the linear Lebesgue measure on the unit circle , It follows that endowed with the norm
becomes a Hilbert space (cf. [6, p. 101].) We will denote it by In fact the map is a unitary map from onto which intertwine the multiplication by on and the multiplication by coordinate function on
We have described three different but unitarily equivalent realization of a bundle shift of rank over the domain We prefer to work with the third realization. It is well known that the harmonic measure on at the point is boundedly mutually absolutely continuous w.r.t the arc length measure on In fact we have
where denote the green function for the domain at the point and denote the directional derivative along the outward normal direction (w.r.t positively oriented ). In this paper, instead of working with harmonic measure on , we will work with arclength measure on This is the approach in Sarason [14]. So, we define the norm of a function in in by
Since the outward normal derivative of the Green’s function is negative on the boundary we have
(2.1) 
where is a positive continuous function on We also see that
where and are the supremum and the infimum of the function on
Hence it is clear that defines an equivalent norm on We let be the Hilbert space which is the same as as a linear space but is given the new norm In fact, the identity map from onto is invertible and intertwines the corresponding multiplication operator by the coordinate function. It is easily verified that the multiplication operator by coordinate function on is also a pure cyclic subnormal operator with spectrum equal to and normal spectrum contained in By a slight abuse of notation, we will denote the multiplication operator by the coordinate function on also by
Using the characterization of all cyclic subnormal operator with spectrum equal to and normal spectrum contained in given by Abrhamse and Douglas, we conclude that for every character the operator on is unitarily equivalent to on for some In the following section we will establish a bijective correspondence (which respects the unitary equivalence class) between these two kinds of bundle shifts. The following Lemma helps in establishing this bijection.
Lemma 2.4.
If be a positive continuous function on then there exist a function in for some character such that almost everywhere (w.r.t arc length measure), on In fact is invertible in the sense that there exist in so that on
Proof.
Since is a positive continuous function on it follows that is continuous on Since the boundary of consists of jordan analytic curves, the Dirichlet problem is solvable with continuous boundary data. Now solving the Dirichlet problem with boundary value we get a harmonic function on with continuous boundary value Let be the multiple value conjugate harmonic function of Let’s denote the period of the multiple valued conjugate harmonic function around the boundary component by
In the above equation negative sign appear since we have assumed that is positively oriented, hence the different components of the boundary except the outer one are oriented in clockwise direction. Now consider the function defined by
Now observe that is a multiplicative holomorphic function on Hence following [17, Lemma 3.6], we have a existence of modulus automorphic function on unit disc so that We find the index of the modulus automorphy for the function in the following way. Around each boundary component along the anticlockwise direction, the value of gets changed by times its initial value. So, the index of is determined by the tuple of numbers given by,
For each of these tuple of numbers, there exist a homomorphism such that these tuple of numbers occur as a image of the generator of the group under the map Also we have Since is continuous on , it follows that is bounded on Hence belongs to with on
The function is also positive and continuous on , as before, there exists a function in with on Since , it easy to verify that index of is exactly and hence is equal to Evidently on ∎
Now we establish the bijective correspondence which preserve the unitary equivalence class, promised earlier. From (2.1), we know that the harmonic measure is of the form for some positive continuous function on Combining this with the preceding Lemma, we see that there is a in with on and a in with on Now consider the map defined by the equation
Clearly, is a unitary operator and its inverse is the operator The multiplication operator intertwines the corresponding operator of multiplication by the coordinate function on the Hilbert spaces and establishing a bijective correspondence of the unitary equivalence classes of bundle shifts. As a consequence we have the following theorem which was proved by Abrahamse and Douglas (See [2, Theorem 5 and 6]) with the harmonic measure instead of the arc length measure
Theorem 2.5.
The bundle shift on is unitarily equivalent to the bundle shift on iff
It can be shown using the result of Abrahamse and Douglas (See [2, Theorem 3]) that for any character , the adjoint of the rank bundle shift lies in Since the bundle shifts is subnormal, it follows that the adjoint of the bundle shifts admits as a spectral set. Consequently, we have an inequality for the curvature of the bundle shifts, namely,
Given any fixed but arbitrary point in in the following section, we recall the proof (slightly different from the original proof given in [11] of the existence of a bundle shift for which equality occurs at in the curvature inequality. However, the main theorem of this paper is the “uniqueness” of such an operator.
Theorem 2.6 (Uniqueness).
If the bundle shift on and the bundle shift on are extremal at the point that is, if they satisfy
then the bundle shifts and are unitarily equivalent, which is the same as
The Hardy space consists of holomorphic function on such that has a harmonic majorant on Each in has a non tangential boundary value almost everywhere. In the usual way is identified with a closed subspace of (see [13, Theorem 3.2]). Let be a positive continuous function on . As the measure and the harmonic measure on are boundedly mutually absolutely continuous one can define an equivalent norm on in the following way
Let denote the linear space endowed with the norm Since the harmonic measure is boundedly mutually absolutely continuous with the arc length measure and is a positive continuous function on it follows that identity map is an invertible map intertwining the associated multiplication operator Thus acquires the structure of a Hilbert space and the operator on it is cyclic, pure subnormal, its spectrum is equal to and finally its normal spectrum is equal to Consequently, the operator on must be unitarily equivalent to the bundle shift on for some character Now, we compute the character
Since is a positive continuous function on using Lemma we have the existence of a character and a function in satisfying on The function is also invertible in the sense that there exist a function in such that on It is straightforward to verify that the linear map defined by
is unitary. Also being a multiplication operator, intertwines the corresponding multiplication operator by the coordinate function on the respective Hilbert spaces. From Lemma it is clear that the character is determined by the following tuple of numbers:
(2.2) 
where is the harmonic function on with continuous boundary value Using this information along with the Theorem 2.5, we deduce the following Lemma which describe the unitary equivalence class of the multiplication operator on
Lemma 2.7.
Let be two positive continuous function on Then the operators on the Hilbert spaces and are unitarily equivalent iff
It also follows from a result of Abrahamse (See [1, Proposition 1.15]) that given a character there exist a invertible element in such that
where are positive constant. Thus we have proved the following theorem.
Theorem 2.8.
Given any character there exists a positive continuous function defined on such that the operator on is unitarily equivalent to the bundle shift on
3. Weighted Kernel and Extremal Operator at a fixed point
Let be a positive continuous function on Since is a reproducing kernel Hilbert space and the norm on is equivalent to the norm on it follows that is also a reproducing kernel Hilbert space. Let denote the kernel function for
The case gives us the Sz̈ego kernel for the domain Associated to the Sz̈ego kernel, there exists a conjugate kernel called the Garabedian kernel, which is related to the Sz̈eego kernel via the following identity.
We recall several well known properties of these two kernels when consists of jordan analytic curves. For each fixed in , the function is holomorphic in a neighbourhood of and is holomorphic in a neighbourhood of with a simple pole at is non vanishing on The function is non vanishing on and has exactly zero in (See [3, Theorem 13.1]). In [12, Theorem 1] Nehari has extended these result for the kernel
Theorem 3.1 (Nehari).
Let be a bounded domain in the complex plane, whose boundary consists of analytic jordan curve and let be a positive continuous function on Then there exist two analytic function and with the following properties: for each fixed in , the function and are holomorphic in is continuous on and is continuous in where denotes a small open disc about ; and are connected by the identity
(3.1) 
These properties determine both functions uniquely.
From we have that The boundary consists of Jordan analytic curves, therefore from the Schwartz reflection principle, it follows that the function and are holomorphic in a neighbourhood of
We have shown that the operator on is unitarily equivalent to a bundle shift of rank . Consequently the adjoint operator lies in admitting as a spectral set from which a curvature inequality follows:
Or equivalently,
Fix a point in The following lemma provides a criterion for the adjoint operator on