GEOMETRIC INTERPOLATION BETWEEN HILBERT SPACES

John E. McCarthy

Abstract. We prove that there is a unique way to construct a geometric scale of Hilbert spaces interpolating between two given spaces. We investigate what properties of operators, other than boundedness, are preserved by interpolation. We show that self-adjointness is, but subnormality and Krein subnormality are not. On the way to this last result, we establish a representation theorem for cyclic Krein subnormal operators.

Introduction

The basic idea of interpolation theory is as follows: one is given a linear operator T with some property P on two different topological vector spaces, X_{0} and X_{1}, and one constructs a family of spaces X_{s} “between” X_{0} and X_{1}, such that T also has property P on each X_{s}, 0 < s < 1. Normally, X_{0} and X_{1} are Banach spaces, and P is the property of being bounded.

One of the first such theorems was proved by M. Riesz [Ri], who showed that if T is bounded from L^{p_{0}}(μ) to L^{p_{0}}(μ) and from L^{p_{1}}(μ) to L^{p_{1}}(μ), then it is bounded from L^{p}(μ) to L^{p}(μ), for all 1 ≤ p_{0} ≤ p ≤ p_{1} ≤ ∞. Since then, several methods have been developed for producing interpolating families X_{s}, the principal ones being the complex method due to Calderón [Ca] and the real method due to Lions and Peetre [LP]. Although these methods always produce interpolation spaces whenever (X_{0}, X_{1}) is a compatible couple, they are useful only in so far as the spaces X_{s} can be concretely realised. Thus Riesz’ theorem above can be proved by showing that for either of these methods, the interpolation spaces are isomorphic to L^{p}(μ); but in general, it is far from clear what the interpolation spaces look like, or whether they are naturally occurring spaces at all.

Typeset by AMS-TEX 2 JOHN E. MCCARTHY

This work was partially supported by NSF grant DMS 9102965.

If one is working with Hilbert spaces (which we always assume to be separable), one would expect the situation to be more transparent, and indeed W. Donoghue gave a classification of all Hilbert spaces that can interpolate exactly between two given Hilbert spaces [Do]. Our first result is that if one makes an additional log-convexity assumption, the interpolating spaces are unique, and there is a canonical way of finding them.

Specifically, let (H_{0}, H_{1}) be a compatible couple of Hilbert spaces, i.e. they both embed continuously in some Hausdorff topological vector space V, and D := H_{0} ∩ H_{1} is dense in both spaces (a typical example of a compatible couple is two spaces of analytic functions on a domain, e.g. the Hardy space and the Bergman space). Let H be another Hilbert space contained in V in which D is dense, so any linear operator defined on H_{0} and H_{1} is at least densely defined on H. Then, for 0 < s < 1, we shall say H is a geometric interpolating space of exponent s between H_{0} and H_{1} if it satisfies the following three properties for linear operators T:

i) If T maps D to D and satisfies ∥T ξ∥_{H_{0}} ≤ λ_{0}∥ξ∥_{H_{0}} and ∥T ξ∥_{H_{1}} ≤ λ_{1}∥ξ∥_{H_{1}}, then ∥T ξ∥_{H} ≤ λ_{0}^{1−s}λ_{1}^{s}∥ξ∥_{H}. ii) If T maps D into some Hilbert space K, and ∥T ξ∥_{K} ≤ λ_{0}∥ξ∥_{H_{0}} and ∥T ξ∥_{K} ≤ λ_{1}∥ξ∥_{H_{1}}, then ∥T ξ∥_{K} ≤ λ_{0}^{1−s}λ_{1}^{s}∥ξ∥_{H}. iii) If T maps some Hilbert space K into D and ∥T ξ∥_{H_{0}} ≤ λ_{0}∥ξ∥_{K} and ∥T ξ∥_{H_{1}} ≤ λ_{1}∥ξ∥_{K}, then ∥T ξ∥_{H} ≤ λ_{0}^{1−s}λ_{1}^{s}∥ξ∥_{K}.

In Section 1 we show that for any compatible couple (H_{0}, H_{1}), a geometric interpolating space of exponent s exists and is unique. In the language of category theory, this says that for each 0 < s < 1 there is a unique functor F_{s} mapping the category of compatible couples of Hilbert spaces to the category of Hilbert spaces, with the property that, for any two compatible couples (H_{0}, H_{1}) and (K_{0}, K_{1}), if T is a linear operator that maps H_{0} to K_{0} with norm λ_{0}, and maps H_{1} to K_{1} with norm λ_{1}, then T maps F_{s}(H_{0}, H_{1}) to F_{s}(K_{0}, K_{1}) with norm less than or equal to λ_{0}^{1−s}λ_{1}^{s}.

In Section 2 we consider what properties other than boundedness are also preserved by geometric interpolation. We believe that this should be a valuable tool in building examples of operators that have certain properties, but not others. Specifically, consider

GEOMETRIC INTERPOLATION BETWEEN HILBERT SPACES 3

the question of the existence of an operator that is polynomially hyponormal but not subnormal. This was open for a long time; recently R. Curto and M. Putinar [CP] have proved that such operators exist, but their proof is non-constructive, and no concrete examples are known. Now suppose interpolation preserved polynomial hyponormality (we cannot, alas, prove this); it does not preserve subnormality (see below), so one would get an easy example of an operator that was polynomially hyponormal but not subnormal.

Although we cannot answer the question of whether hyponormality is preserved by interpolation, we can answer the question, subject to one space being contained in the other, for normality (yes), subnormality (no) and Krein subnormality (no). On the way to the last result, we establish a representation theorem for cyclic Krein subnormal operators.

Section 1 : Geometric Interpolation

Let (H_{0}, H_{1}) be a compatible couple of Hilbert spaces. Because the inner product for H_{1}, (., .)_{1} is a Hermitian form on a dense subspace of H_{0}, there is a positive (not necessarily bounded) operator A on H_{0} such that, for any ξ, η in H_{0} ∩ H_{1}, (ξ, η)_{1} = (ξ, Aη)_{0}. For 0 < s < 1, define a new inner product on H_{0} ∩ H_{1} by (ξ, η)_{s} = (ξ, A^{s}η)_{0}. The closure of H_{0} ∩ H_{1} with respect to the norm given by the inner product we will call H_{s}.

We remark that using powers of a positive operator to define interpolation spaces seems to have first appeared in a paper by J.L. Lions [Li], where, under the assumption that one of the Hilbert spaces is contained in the other, he proves essentially that the norm of an operator on an intermediate space will be bounded by some constant times the weighted geometric mean of the norms on the end-spaces. That the constants can be chosen to be one (again assuming that one Hilbert space contains the other) was proved by Krein and Petunin [KP] as part of their general theory of scales of Banach spaces (letting s range over ℝ gives a natural scale of Hilbert spaces); see also the book by Krein, Petunin and Semenov [KPS].

Thus the fact that H_{s} is a geometric interpolating space is really already known; the significance of our proof is that it is simple and short, and that the operator theoretic approach naturally leads to the (new) result that the space is unique. This unicity, and

4 JOHN E. MCCARTHY

the canonical construction of the spaces, make the geometric interpolating spaces natural objects of study for operator theory.

We note that the proof of existence is based on an idea of P. Halmos [Ha].

Theorem 1.1. Let (H_{0}, H_{1}) and (K_{0}, K_{1}) be compatible couples of Hilbert spaces. Suppose T is a linear operator that maps H_{0} to K_{0} with norm λ_{0}, and maps H_{1} to K_{1} with norm λ_{1}. Then T maps H_{s} to K_{s} with norm less than or equal to λ_{0}^{1−s}λ_{1}^{s}. Moreover, H_{s} is the unique geometric interpolation space of exponent s between H_{0} and H_{1}.

Proof.

(Existence) Let A be the positive operator on H_{0} that gives the H_{1} inner product, and B be the positive operator on K_{0} that gives the K_{1} inner product. First, suppose A^{−1} and B are bounded, and that s = \frac{1}{2}. Then we have

∥T ∥_{H_{0}→K_{0}} = λ_{0}
∥B^{\frac{1}{2}} T A^{− \frac{1}{2}} ∥_{H_{0}→K_{0}} = λ_{1}.

The norm of T from H_{\frac{1}{2}} to K_{\frac{1}{2}} equals the norm of S := B^{\frac{1}{4}} T A^{− \frac{1}{4}} from H_{0} to K_{0}. Letting ρ denote the spectral radius, we have

∥S∥^{2} = ∥S^{∗}S∥ = ρ(S^{∗}S)
= ρ(A^{\frac{1}{4}} S^{∗}SA^{− \frac{1}{4}} )
≤ ∥A^{\frac{1}{4}} S^{∗}SA^{− \frac{1}{4}} ∥ ≤ ∥T^{∗}∥_{K_{0}→H_{0}}∥B^{\frac{1}{2}} T A^{− \frac{1}{2}} ∥_{H_{0}→K_{0}}
≤ λ_{0}λ_{1}.

So the theorem is true for s = \frac{1}{2}. Now interpolating between H_{0} and H_{\frac{1}{2}} gives the theorem for s = \frac{1}{4}, and interpolating between H_{\frac{1}{2}} and H_{1} gives the theorem for s = \frac{3}{4}. Similarly, the theorem holds for any dyadic rational in [0, 1], and, by continuity of ∥B^{s/2}T A^{−s/2}∥, any real s in [0, 1].

The assumption that A^{−1} and B be bounded can now be dropped by approximating them by the “truncated” operators A_{n}^{−1} := ∫_{0}^{n} tdE_{A^{−1}}(t) and B_{n} := ∫_{0}^{n} tdE_{B}(t), where

GEOMETRIC INTERPOLATION BETWEEN HILBERT SPACES 5

E_{A^{−1}}(.) and E_{B}(.) are the respective spectral projections, for it is easy to check that ∥B^{s/2}T A^{−s/2}∥ = lim_{n→∞} ∥B_{n}^{s/2}T A_{n}^{−s/2}∥.

(Uniqueness) Let G be another geometric interpolation space. Let E(.) be the spectral measure of A. Fix b > a > 0; let K_{ab} be the space E[a, b)H_{0} with the H_{0} norm. Let T be the operator of orthogonal projection onto K_{ab}, and R be the inclusion of K_{ab} into H_{0} ∩ H_{1}. Then the norm of T on G is at most \frac{1}{a^{(1−s)/2}} , and the norm of R into G is at most b^{(1−s)/2}. So it follows that

a^{(1−s)/2}∥ξ∥_{H_{0}} ≤ ∥ξ∥_{G} ≤ b^{(1−s)/2}∥_{H_{0}}.  (1.2)

If we can show that whenever [a_{0}, b_{0}) ∩ [a_{1}, b_{1}) is empty, E[a_{0}, b_{0})H_{0} and E[a_{1}, b_{1})H_{0} are orthogonal in G, then (1.2) yields, as a tends to b, that the G-norm and the H_{s}-norm coincide. But those two subspaces must be orthogonal, or else the projection E[a_{0}, b_{0}), of norm one on both H_{0} and H_{1}, would have norm greater than one on G. □

We remark that Calderón’s complex method of interpolation also yields a geometric interpolation space, so by uniqueness it must be the same space as above.

Section 2 : Normality and Subnormality

Interpolation theory started as an effort to prove operators were bounded on a range of spaces, knowing how to prove this directly on the end-spaces. But it is of interest to ask what other properties of linear operators are preserved by geometric interpolation. Calderón showed that compactness is preserved [Ca]; we now prove that so is normality, provided one space is actually contained in the other (we don’t believe this stipulation is actually necessary, but we cannot prove the result in general).

For the rest of the paper, (H_{0}, H_{1}) will be a compatible couple of Hilbert spaces, and A will be the (perhaps densely defined) positive operator satisfying (ξ, η)_{H_{1}} = (A^{1/2}ξ, A^{1/2}η)_{H_{0}} for all ξ, η in H_{0} ∩ H_{1}. The geometric interpolating space H_{s} will be as in the previous section, and s will always lie in the interval [0, 1]. Note that A is bounded if and only if H_{0} is contained in H_{1}.

6 JOHN E. MCCARTHY

Theorem 2.1. Suppose H_{0} is contained in H_{1}, and the operator N : H_{0} → H_{0} is normal on both H_{0} and H_{1}. Then N is normal on H_{s} for all s.

Proof. The operator N : H_{1} → H_{1} is unitarily equivalent to the operator M := A^{1/2}NA^{−1/2} on H_{0}. So M and N are normal operators, satisfying MA^{1/2} = A^{1/2}N. By the Fuglede-Putnam theorem - see Conway [Co, p.81], we also have M^{∗}A^{1/2} = A^{1/2}N^{∗}, and taking adjoints gives A^{1/2}M = NA^{1/2}. Therefore

AN = A^{1/2}MA^{1/2} = NA^{1/2}A^{1/2} = NA.

So N commutes with A, and hence all powers of A. Therefore on each H_{s}, the operator N is unitarily equivalent to N on H_{0}, and hence is normal. □

Note that if one assumes that N is actually self-adjoint on both H_{0} and H_{1}, one does not have to appeal to the Fuglede-Putnam theorem, so there is no restriction that A be bounded above or below.

An operator is subnormal if it is the restriction of a normal operator to an invariant subspace. If μ is a compactly supported measure on ℂ, and P^{2}(μ) is the closure of the polynomials in L^{2}(μ), multiplication by the independent variable on P^{2}(μ) is a cyclic subnormal operator, denoted S_{μ}; moreover all cyclic subnormal operators arise this way [Co]. The domain of analyticity of P^{2}(μ) is the largest open set U ⊂ ℂ such that if a sequence of polynomials converges in norm in P^{2}(μ), it converges uniformly on compact subsets of U. It then makes sense to assign values to elements of P^{2}(μ) at points of U. Moreover, if P^{2}(μ) has no L^{2}-summand (equivalently, S_{μ} has no reducing subspace on which it is normal), then an element of P^{2}(μ) is uniquely determined by its values on U, and P^{2}(μ) can be thought of as a space of analytic functions on U - Thomson [Th]. So given two pure spaces P^{2}(μ) and P^{2}(ν), with each component of the domain of analyticity of one intersecting the domain of analyticity of the other, they form a compatible couple (embedded in the space of analytic functions on the intersection of their respective domains). One can ask what the interpolation spaces look like, though they seem somewhat inaccessible in general. A special case was considered in [CMcCW].

GEOMETRIC INTERPOLATION BETWEEN HILBERT SPACES 7

Let us look, however, at radial measures, dμ(re^{iθ}) = \frac{1}{2π} dτ(r)dθ. Then \frac{1}{2π} ∫ z^{n}\overline{z^{m}}dτ(r)dθ = δ_{nm}α_{2n} where {α_{n}} is the moment sequence of τ (let us call τ the radial part of μ). The domain of analyticity is a disk centered at 0. Suppose ν is another radial measure, and let β_{n} be the moments of its radial part. Then the interpolating space of exponent s between P^{2}(μ) and P^{2}(ν) is again a space of analytic functions, in which the inner product of two polynomials ∑ a_{n}z^{n} and ∑ b_{m}z^{m} is ∑ a_{n}\overline{b}_{n}(α_{2n})^{1−s}(β_{2n})^{s}. The operator of multiplication by z on this interpolation space is subnormal if and only if (α_{2n})^{1−s}(β_{2n})^{s} are the even moments of some measure compactly supported in ℝ^{+}.

If we take all the β_{n}’s to be 1, then P^{2}(ν) is the Hardy space H^{2}, and S_{ν} is the unilateral shift. If, for all 0 < s < 1, {α_{n}^{s}} is a moment sequence, the measure τ is called infinitely divisible (notice that, for a measure supported on [0, L], the even moments determine the odd moments). Thus we have:

Theorem 2.2. Let μ be a radial measure, with radial part τ. Let H_{0} = P^{2}(μ), and H_{1} = H^{2}. Then the operator of multiplication by z is subnormal on every interpolating space H_{s} if and only if τ is infinitely divisible.

Infinitely divisible measures have been studied by R.A. Horn [Ho].

Example 2.3. Let τ be Lebesgue measure on [0, 1]. This is infinitely divisible, as can be checked by Hausdorff’s moment theorem. So is the measure “rdr”, so interpolating between the Bergman shift and the Hardy shift always yields a subnormal operator.

Example 2.4. Now let τ have an atom at \frac{1}{2} and an atom at 1, both of weight 1. Then α_{n} = 1+ \frac{1}{2^{n}}. The sequence \{\sqrt{α_{n}}\} is not a moment sequence, as the 3-by-3 Hankel matrix

( \begin{matrix}\sqrt{2} & \sqrt{\frac{3}{2}} & \sqrt{\frac{5}{4}} \\ \sqrt{\frac{3}{2}} & \sqrt{\frac{5}{4}} & \sqrt{\frac{9}{8}} \\ \sqrt{\frac{5}{4}} & \sqrt{\frac{9}{8}} & \sqrt{\frac{17}{16}} \end{matrix} )

is not positive. Thus we have an example of interpolating between two subnormal operators to get an operator that is not subnormal (in the language of Curto and Putinar, it is not even 2-hyponormal). As we interpolate between H_{0} and H_{1}, we get a family of weighted

8 JOHN E. MCCARTHY

shifts given by

W e_{n} = (\frac{4^{n+1} + 1}{4^{n+1} + 4})^{\frac{1−s}{2}} e_{n+1}.

For s equal to 0 or 1, these shifts are subnormal. Are they polynomially hyponormal for all s? If so, they would provide the first example of an operator that is polynomially hyponormal but not subnormal (the existence of such operators has recently been proved by Curto and Putinar [CP]).

Section 3 : Krein Subnormality

A Krein space is a Hilbert space K, (., .) with an additional indefinite inner product < ., . > defined, where this indefinite form is given in terms of a symmetry (a self-adjoint unitary) J by

< x, y >= (x, Jy).

Recently, operators on Krein spaces have attracted much attention - see e.g. the book by Azizov and Iokhvidov [AI] and the review of it by Rodman [Ro]. C. Cowen and S. Li initiated the study of Krein subnormal operators and their connection with moment problems [CL].

A linear operator T_{1} on a Krein space K_{1} is a Krein extension of the operator T_{0} on K_{0} if

(i) K_{0} is a closed subspace of K_{1}; (ii) < x, y >_{0}=< x, y >_{1} for all x, y in K_{0}; and (iii) T_{0}x = T_{1}x for all x in K_{0}. There is a natural decomposition K = K^{+} ⊕ K^{−} into the positive and negative eigenspaces of J. A linear operator T on K is called fundamentally reducible if it leaves these spaces invariant (and in this case the adjoint of T is the same with respect to either inner product). An operator on a Hilbert space H is called Krein subnormal if it has a continuous fundamentally reducible Krein extension that is normal (i.e. commutes with its adjoint). Note that H need not be contained in K^{+}, it can be at an angle to it; but it can’t intersect K^{−} because the inner product on H is positive. Let us also remark that Wu J. has studied

GEOMETRIC INTERPOLATION BETWEEN HILBERT SPACES 9

a weaker form of subnormality, in which K_{0} is not required to be closed in K_{1}, and the extension need not be fundamentally reducible; he proved that every bounded operator is subnormal in this weaker sense [Wu].

Let μ = μ_{+} − μ_{−} be a real measure. Then L^{2}(|μ|) is a Krein space with inner product

< f, g >= ∫ f\overline{g}dμ_{+} − ∫ f\overline{g}dμ_{−}.

Call this space K^{2}(μ). Just as all cyclic subnormal operators can be represented as multiplication by z on P^{2}(ν) for some positive measure ν, Cowen and Li proved that all cyclic Krein subnormal operators can be represented as multiplication by z on some Q^{2}(μ), the closure of the polynomials in K^{2}(μ) [CL, Theorem 9]. However, not all real measures μ can be obtained; here is a characterization of those that can.

Theorem 3.1. Let μ have Jordan decomposition μ_{+} − μ_{−}. Multiplication by the independent variable on Q^{2}(μ) is Krein subnormal if and only if there exists some constant c < 1 such that, for all polynomials p,

∫ |p|^{2}dμ_{−} ≤ c ∫ |p|^{2}dμ_{+}.  (3.2)

Proof. (⇒) By [CL, Theorem 5] the ‘forgetful map’ from K^{2}(μ) to L^{2}(|μ|) that forgets about the Krein space structure implements a similarity between M_{z} on Q^{2}(μ) and M_{z} on P^{2}(|μ|). Therefore there is some constant c_{1} such that

∫ |p|^{2}d|μ| ≤ c_{1} ∫ |p|^{2}dμ,

and so

∫ |p|^{2}dμ_{−} ≤ \frac{c_{1} − 1}{c_{1} + 1} ∫ |p|^{2}dμ_{+}.

(⇐) Multiplication by z on K^{2}(μ) is clearly a normal fundamentally reducible Krein extension. So all that remains is to prove that Q^{2}(μ) is a Hilbert space, i.e. that the sesquilinear form < f, g >= ∫ f\overline{g}dμ is positive definite. So suppose p_{n} are polynomials

10 JOHN E. MCCARTHY

that converge to some non-zero element f of Q^{2}(μ). Then p_{n} converges to f in L^{2}(|μ|), so |p_{n}|^{2} converges to |f|^{2} in L^{1}(|μ|), and hence

∫ f\overline{f}dμ = lim_{n→∞}(∫ |p_{n}|^{2}dμ_{+} − ∫ |p_{n}|^{2}dμ_{−})

≥ (1 − c) lim_{n→∞} ∫ |p_{n}|^{2}dμ_{+}

≥ (1 − c) ∫ |f|^{2}dμ_{+}

≥ \frac{1 − c}{1 + c} ∫ |f|^{2}d|μ|

> 0.

□

Note that if μ_{−} is non-zero, the operator cannot be subnormal, because if a measure ν satisfies ∫ |p|^{2}dν = ∫ |p|^{2}dμ for all polynomials p, then ν = μ.

If we take μ_{+} to be normalized area measure on the unit disk, and μ_{−} to be an atom of weight α at zero, we get a Krein subnormal weighted shift for α < 1. Direct calculation shows that this operator is hyponormal if α ≤ 1/4. This shows that the answer to Question 4 of [CL], whether a hyponormal Krein subnormal operator must be subnormal, is no (Cowen and Li say in a note added in proof that they have also found an example).

Obviously a theorem similar to 2.2 holds if one wants interpolation between H^{2} and Q^{2}(μ) to preserve Krein subnormality for multiplication by z - the powers of the moments of the radial part of μ must themselves be moments of the radial part of a measure satisfying (3.2). That this does not always hold is shown by the following example. Example 3.3. Let λ be linear measure on the multiplicative semi-group [0, 1], and let ∗ denote the convolution of measures with respect to the semi-group. Let x = 0.58 (which is slightly greater than 1/\sqrt{3}). Let τ = λ + δ_{0} − δ_{x}, where δ_{x} is a point mass at x, and let σ = τ ∗ τ.

Claim: The measure μ given by dμ(z) = \frac{1}{2π} dσ(r)dθ satisfies (3.2) for c = .99996. Proof: We must show that for any polynomial p(z) = ∑_{n=0}^{N} a_{n}z^{n},

∫ |p|^{2}d(2δ_{x} ∗ λ + δ_{0})(r)dθ ≤ c ∫ |p|^{2}d(λ ∗ λ + 2λ ∗ δ_{0} + δ_{x^{2}})(r)dθ.

As the moments of τ are \frac{1}{n+1} − x^{n} for n > 0, and 1 for n = 0, and because the moments of σ are the squares of the moments of τ, this is equivalent to:

3|a_{0}|^{2} + ∑_{n=1}^{N} |a_{n}|^{2}(\frac{2x^{2n}}{2n + 1}) ≤ c[4|a_{0}|^{2} + ∑_{n=1}^{N} |a_{n}|^{2}(x^{4n} + \frac{1}{(2n + 1)^{2}} )].

The verification of this inequality is elementary.

Claim: The measure ν given by dν(z) = \frac{1}{2π} dτ(r)dθ does not satisfy (3.2) for any c ≤ 1. Proof: Let p(z) = z. Then ∫ |z|^{2}dν_{+} = \frac{1}{3}, which is smaller than ∫ |z|^{2}dν_{−} = x^{2}.

Claim: Let H_{0} = Q^{2}(μ) and H_{1} = H^{2}. Then the operator of multiplication by z is Krein subnormal on H_{0} and H_{1} but not on H_{\frac{1}{2}} . Deny. Then there would be a radial real measure κ with radial part ρ satisfying

∥z^{n}∥_{H_{\frac{1}{2}}}^{2} = ∫_{[0,1]} r^{2n}dρ(r)
= \sqrt{∥z^{n}∥_{H_{0}}^{2}}
= \sqrt{∫_{[0,1]} r^{2n}dσ(r)}
= | ∫_{[0,1]} r^{2n}dτ(r)|.

But all of τ’s moments, except for the first and second, are positive. So ρ must equal τ, and hence κ equals ν and does not satisfy (3.2). □

References

[AI] T.Ya. Azizov and I.S. Iokhvidov “Linear operators in spaces with an indefinite metric,” Wiley, New York, 1989 [BL] J.Bergh and J.Löfström “Interpolation spaces,” Springer-Verlag, Berlin,1976 [Ca] A.P. Calderón “Intermediate spaces and interpolation, the complex method,” Studia

Math.  24  [1964]  113-190
[Co] J.B. Conway “Subnormal Operators,”  Pitman,  Boston,  1981

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[CL] C. Cowen and S. Li “Hilbert space operators that are subnormal in the Krein space sense,” J. Operator Theory 20 [1988] 165-181 [CMcCW] M. Cwikel, J.E. McCarthy and T. Wolff “Interpolation between weighted Hardy spaces,” to appear in Proc. AMS [CP] R. Curto and M. Putinar “Nearly subnormal operators and moment problems,” to appear

[Do] W.F. Donoghue “The interpolation of quadratic norms,” Acta Math. 118 (1967) 251-270

[Ha] P. Halmos “Quadratic interpolation,” J. Operator Theory 7 [1982] 303-305 [Ho] R.A. Horn “Infinitely divisible positive definite sequences,” Trans. A.M.S. 136 (1969) 287-303

[KP] S.G. Krein and Yu.I. Petunin “Scales of Banach spaces,” Russian Math. Surveys 21 no.2 [1966] 85-159 [KPS] S.G. Krein, Yu.I. Petunin and E.M. Semenov “Interpolation of linear operators,” Translations of mathematical monographs, Volume 54, American Mathematical Society, Providence

[Li] J.L. Lions “Espaces intermédiaires entre éspaces Hilbertiens et applications,” Bull. Math. de la Soc. Sci. Math. Phys. de la R.P.R. Tome 2 (50) no. 4 [1958] 419-432 [LP] J.L. Lions and J. Peetre “Sur une classe d’éspaces d’interpolation,” Inst. Hautes Etudes Sci. Publ. Math. 19 [1964] 5-68 [Ri] M. Riesz “Sur les maxima des formes bilinéares et sur les fonctionelles linéaires,” Acta

Math.  29  (1926)  465-497
[Ro] L. Rodman:  Review of [AI], Bull.  A.M.S. 25 (1991) 111-123

[Th] J.E. Thomson “Approximation in the mean by polynomials” Annals of Math. 133 (1991) 477-507 [Wu] Wu Jingbo “Normal extensions of operators to Krein spaces,” Chinese Ann. Math. 8B (1987) 36-42

Department of Mathematics Washington University St. Louis MO 63130

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U.S.A. mccarthy@jezebel.wustl.edu