THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the S-lemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry.

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A Survey of the S-Lemma

Functions of one complex variable II

Overview Preliminaries The Cauchy transform as a function The Cauchy transform as an operator Topologies on the space of Cauchy transforms Which functions are Cauchy integrals? Multipliers and divisors The distribution function for Cauchy transforms The backward shift on $H^2$ Clark measures The normalized Cauchy transform Other operators on the Cauchy transforms List of symbols Bibliography Index.

The Cauchy Transform

Let ${\bold D}$ denote the unit disk in the complex plane and let $m$ be the area Lebesgue measure on ${\bold D}$. Given a positive integrable function $w$ (a weight) on ${\bold D}$, let $L^p_{\rm a}(w)$ denote the collection of analytic functions $f$ on ${\bold D}$ such that $|f|^pw$ is integrable. Given an analytic function $g$ on ${\bold D}$, the operator $T_g$ is defined by $T_g f(z) = \int_0^{z} f(\zeta)g'(\zeta)\,d\zeta$. The authors consider conditions on $g$ such that $T_g$ is bounded on $L^p_{\rm a}(w)$. In many cases the derivative $D$ is an isomorphism between the subspace of $L^p_{\rm a}(w)$ consisting of functions vanishing at $0$ and some space $L^p_{\rm a}(v)$ with another weight $v$. Thus, the question of boundedness or compactness of $T_g$ becomes the corresponding question for the operator of multiplication by $g'$ acting from $L^p_{\rm a}(w)$ to $L^p_{\rm a}(v)$. The authors consider only $w$ which are radial: $w(re^{i\theta}) = w(r)$. In the first part of the paper it is shown that $\int |f|^pw\,dm \le C\int |f'(z)|^p v(|z|) \,dm(z)$, $p \ge 1$, where $v(r) = \int_r^1 w(u)\,du$. Under the assumption that $v(r) \le C(1 - r)w(r)$, which is valid in particular for $w(r) \equiv (1 - r)^\alpha$, $\alpha > -1$, it follows that $T_g$ is bounded when $g'(z)(1 - |z|)$ is bounded. The converse can be proved in a more general setting. It is obtained by estimating the norm of the linear functional $D_\lambda\colon f \mapsto f'(\lambda)$ in terms of that of the evaluation functional $L_\lambda\colon f \mapsto f(\lambda)$. Rather general hypotheses on a Banach space of analytic functions are obtained in order that $\| D_\lambda \|(1 - |\lambda|) \le C\| L_\lambda \|$. This leads immediately to the converse: if the operator $T_g$ is bounded, then $g'(z)(1 - |z|)$ is bounded. Several classes of weights are shown to satisfy the hypotheses required for both the necessity and the sufficiency of the condition. In all cases, the problem of compactness of $T_g$ is also considered, and the solution involves the little-oh versions of the same conditions for boundedness. In the special case where $p = 2$ and $w(r) = (1 - r)^\alpha$ the Schatten class of $T_g$ is determined. The techniques are briefly applied to the weight $w(r) = \exp[ -\beta(1 - r)^{-\alpha}]$, $\alpha > 0$. In this case $v(r) \le C(1 - r)^{\alpha + 1}w(r)$, and that leads to the conclusion that $T_g$ is bounded when $g'(z)(1 - |z|)^{\alpha + 1}$ is bounded. The necessity of this condition is left open. A theorem of V. L. Oleinik [Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 47 (1974), 120--137, 187, 192--193; MR0369705 (51 #5937) (Theorem 3.3)] shows that it is necessary at least when $\alpha > 1$. (Less)

Integration operators on Bergman spaces

A celebrated theorem in operator theory is A. Beurling's description of the invariant subspaces in $H^2$ in terms of inner functions [Acta Math. {\bf81} (1949), 239--255; MR0027954 (10,381e)]. To do the same thing for the Bergman space $L^2_a$ has been deemed virtually impossible by many analysts, in view of the fact that the lattice of invariant subspaces is so large, and that the invariant subspaces may have weird properties as viewed from the $H^2$ perspective. The size of the lattice can be appreciated from the known fact that essentially every operator on separable Hilbert space can be realized as the compression of the Bergman shift on $M\ominus N$, where $M$ and $N$ are invariant subspaces, $N\subset M$. 

But a Beurling-type theorem is precisely what the present paper delivers. Given an invariant subspace $M$ in $L^2_a$, consider the subspace $M\ominus TM$, where $T$ stands for multiplication by $z$. This makes sense because $TM$ is a closed subspace of $M$. In Beurling's $H^2$ case, $M\ominus TM$ is one-dimensional and spanned by an inner function. In the $L^2_a$ setting, the dimension of $M\ominus TM$ may be arbitrarily large, even infinite. However, with the correct analogous definition of inner functions in $L^2_a$, all vectors of unit norm in\break $M\ominus TM$ are $L^2_a$-inner. Following Halmos, the subspace $M\ominus TM$ is called the wandering subspace of $M$. 

Given an invariant subspace, a natural question is: which collections of elements generate it? In particular, one can ask for the least number of elements in a set of generators. It is known that the dimension of the wandering subspace represents a lower bound for the least number of generators. In the paper, it is shown that any orthonormal basis in the wandering subspace (which then consists of $L^2_a$-inner functions) generates $M$ as an invariant subspace. This settles the issue of the minimal number of generators. 

Let $P$ be the orthogonal projection $M\to M\ominus TM$, and let $L\colon M\to M$ be the operator such that $TL$ is the orthogonal projection $M\to TM$. Then, for $f\in M$, $f=Pf+TLf$. If we do the same for $Lf\in M$, we get $Lf=PLf+TL^2f$ and, inserting it into the original relation for $f$, we get $f=Pf+TPLf+T^2L^2f$. As we go on repeating this process, we get $f=Pf+TPLf+T^2PL^2f+\cdots+T^{n-1}PL^{n-1}f+T^nL^nf$. The point with this decomposition is that, apart from the last term, each term is of the form $T$ to some power times an element of $M\ominus TM$, so that $Pf+TPLf+T^2PL^2f+\cdots+T^{n-1}PL^{n-1}f$ is in\break $[M\ominus TM]$, the invariant subspace generated by $M\ominus TM$. If the operators $T^nL^n$ happened to be uniformly bounded, as they are in the case of $H^2$, $T^nL^nf$ would tend to $0$ in the weak topology, and $f$ would be in the weak closure of $[M\ominus TM]$, which by standard functional analysis coincides with $[M\ominus TM]$. However, for the Bergman space, it seems unlikely that the $T^nL^n$ are uniformly bounded for all possible invariant subspaces $M$, although no immediate counterexample comes to mind. 

For this reason, the authors try Abel summation instead, and consider for $0<s<1$ the operators $R_s\colon M\to [M\ominus TM]$ given by $R_s=P+sTPL+s^2T^2PL^2+s^3T^3PL^3+\cdots$; it is easy\break to show that the series converges in norm. To prove that $M=\break [M \ominus TM]$, it suffices to check that (a) $\|R_s\|\le C$ for some constant independent of $s$, and (b) $R_sf\to f$ in the topology of uniform convergence on compacts in the unit disk. This is so because $R_sf\in [M\ominus TM]$ then converges weakly to $f\in M$ as $s\to1$. Part (b) is easy; the trick is to obtain (a). The authors show at an early stage that for $\lambda\in\bold D$, $M=(M\ominus TM)+\break (T-\lambda)M$, and that the sum is direct (in the Banach space sense). They write $Q_\lambda f$ for the skewed projection operator $M\to M\ominus TM$ associated with this decomposition. The operator $Q_\lambda$ has a convergent Taylor series expansion $Q_\lambda=P+\lambda PL+\lambda^2 PL^2+\lambda^3PL^3+\cdots$, which very much resembles the expression for $R_s$. In fact, one sees that $R_sf(w)=Q_{sw}f(w)$. Now one observes that $Q_\lambda f(\lambda)=f(\lambda)$, because the element of $(T-\lambda)M$ which one has to add to $Q_\lambda f$ to obtain $f$ vanishes at the point $\lambda$. It follows that $R_sf(w)\to f(w)$ normally in $\bold D$ as $s\to1$. 

We come to the hard part: obtaining the uniform boundedness of the operators $R_s$. What is needed is a concrete representation formula for the norm in $M$ which makes it possible to compare the norms of $f$ and $R_sf$. Here, the factorization theory due to the reviewer [J. Reine Angew. Math. 422 (1991), 45--68; MR1133317 (93c:30053)], and subsequent improvements by Duren-Khavinson-Shapiro-Sundberg, come to assistance. Let $\varphi$ be an $L^2_a$-inner function, and let $f\in[\varphi]$, the invariant subspace generated by $\varphi$. The norm of $f$ can then be expressed in terms of the quotient $f/\varphi$. That formula suggests the norm identity (1) $\|f\|^2=\pi^{-1}\int_\bold D\|Q_w f\|^2dA(w)+\pi^{-2}\int_{\bold D\times\bold D}\Gamma(z,w) \Delta_z\Delta_w|Q_wf(z)|^2\,dA(z)\,dA(w)$, which turns out to be valid for general invariant subspaces $M$, where $\Gamma(z,w)$ is the biharmonic Green function for the unit disk (suitably normalized), which is known to be positive. One first checks that the formula (1) holds for all $f$ in $[M\ominus TM]$. Second, a related integral formula for the norm in $M$, in terms of integrals along concentric circles, is established for all $f\in M$. Then an intricate argument, involving Green's formula and rather subtle analysis of signs of functions, shows that for $f\in M$, we have at least a $\ge$ inequality in (1). To get the identity (1) for general $f\in M$, it is necessary to show first that $M=[M\ominus TM]$. 

Formula (1), which we know to hold for $f\in [M\ominus TM]$ and with $\ge$ for general $f\in M$, applies to $R_s f\in[M\ominus TM]$ with equality, and if we notice that $Q_wR_sf(z)=Q_{sw}f(z)$, we get (2) $\|R_sf\|^2= \pi^{-1}\int_\bold D\|Q_{sw} f\|^2dA(w)+\pi^{-2}\int_{\bold D\times\bold D} \Gamma(z,w)\Delta_z\Delta_w|Q_{sw}f(z)|^2\,dA(z)\,dA(w)$. An almost radial monotonicity property of $\Gamma(z,w)$ in the $w$ variable then shows that the last term on the right-hand side of (2) has a $\limsup$ as $s\to1$ bounded by twice the value one gets when $s$ is set equal $1$. Moreover, the first term on the right-hand side of (2) has a $\limsup$ as $s\to 1$ which equals what one gets when $s=1$ is plugged in. Thus, $\limsup_s\|R_sf\|\le 2\|R_1f\|\le2\|f\|$. It follows that $R_sf\to f$ weakly, which completes the proof. In the paper, it is even shown that $R_sf\to f$ in norm as $s\to1$. Incidentally, the proof also answers affirmatively one of the conjectures raised by the reviewer concerning polynomial approximation in certain weighted Bergman spaces [in Linear and complex analysis. Problem book 3. Part II, 114, Lecture Notes in Math., 1574, Springer, Berlin, 1994]. 

The above-mentioned theorem is new also in the case when the wandering subspace $M\ominus TM$ is one-dimensional. 

The theorem seems to represent a breakthrough in our understanding of the invariant subspaces of the Bergman space. What is desirable and remains to be developed is a better understanding of the wandering subspaces.

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Beurling's Theorem for the Bergman space

An integral operator on $Hsp p$ and Hardy's inequality.

Let g be an analytic function on the unit disk D . We study the operator on the Hardy spaces Hp . We show that Tg is bounded on Hp , 1 ≤ p < ∞ it and only if g ∊ BMOA and compact if and only if g ∊ VOMA. Further on the Hilbert space H2 Tg is in the Schattcn ρ-class it and only if g is in the Besov space Bp , 1 < p < ∞. A relation of Tg with integration operators is also discussed.

Alexandru Aleman

Papers

Integration operators on Bergman spaces

Beurling's Theorem for the Bergman space

An integral operator on $Hsp p$ and Hardy's inequality.

An integral operator on $H\sp p$

Spectra of Integration Operators on Weighted Bergman Spaces