S. ROLEWICZ

ON THE ISOMORPHISM AND APPROXIMATIVE DIMENSION OF SPACES OF HOLOMORPHIC FUNCTIONS

(Presented by Academician A. N. Kolmogorov on 16 IV 1960)

In this note we present certain results concerning the isomorphic classification and the classification by approximative dimension of various spaces of holomorphic functions of many variables.

Let \(X, Y\) be linear-metric spaces. The spaces \(X\) and \(Y\) are called isomorphic if there exists a linear operator \(U\) mapping \(X\) one-to-one onto the space \(Y\), continuous together with its inverse.

Let \(X\) be a linear-metric space. Denote by \(\Phi(X)\) the class of functions \(\varphi(\varepsilon)\) such that, for every compact set \(K\) and every neighborhood of zero \(U\), there exists a number \(\varepsilon_0>0\) such that for \(\varepsilon<\varepsilon_0\) there exist \(N\leqslant \varphi(\varepsilon)\) points \(x_1,\ldots,x_N\) such that*

\[ K \subset \bigcup_{n=1}^{N}(x_n+\varepsilon U). \]

The spaces \(X\) and \(Y\) have equal approximative dimension, \(d_a(X)=d_a(Y)\), if \(\Phi(X)=\Phi(Y)\). The space \(X\) has approximative dimension no greater than that of the space \(Y\), \(d_a(X)\leqslant d_a(Y)\), if \(\Phi(X)\supset \Phi(Y)\). The approximative dimension of the space \(X\) is less than the approximative dimension of the space \(Y\), \(d_a(X)<d_a(Y)\), if \(d_a(X)\leqslant d_a(Y)\) and \(d_a(X)\ne d_a(Y)\). If the spaces \(X\) and \(Y\) are isomorphic, then \(d_a(X)=d_a(Y)\). If \(X\) is a subspace of \(Y\), then \(d_a(X)\leqslant d_a(Y)\). The concept of approximative dimension belongs to A. N. Kolmogorov \((^5)\).

Let \(D\) be a domain of \(k\)-dimensional complex space. By \(\hat H(D)\) we denote the space of all holomorphic functions \(x(z_1,\ldots,z_k)\) defined in the domain \(D\), with the usual operations of addition and multiplication by a complex number and with the locally convex topology induced by all seminorms

\[ \|x\|_F=\sup_{(z_1,\ldots,z_k)\in F}|x(z_1,\ldots,z_k)|, \]

where \(F\) is an arbitrary compact set contained in \(D\). There exists a metric equivalent to this topology such that the space is a complete linear-metric space. Convergence of a sequence in the space \(\hat H(D)\) is equivalent to almost uniform convergence. For every \(D\) the space \(\hat H(D)\) is a nuclear space.**

In this note we investigate isomorphisms and equalities or inequalities of the approximative dimension of the space \(H(D)\) for various classes of domains \(D\). In addition, for some classes of domains \(D\) there is given

* \((x_n+\varepsilon U)\) denotes the set of all elements of the form \(x_n+\varepsilon y\), where \(y\in U\).

** The definition and properties of nuclear spaces are given in \((^4)\); see also \((^9)\).

an isomorphic representation of the space \(H(D)\) by means of Köthe sequence spaces. The note contains a generalization of some results of the works \((^{4-6,8})\).

A domain \(D\) is called a polycylinder if it is the Cartesian product of one-dimensional domains \(D_i,\ i=1,2,\ldots,k\), \(D=D_1\times D_2\times\cdots\times D_k\). We shall denote by \(C\) the complex plane, by \(C_0\) the unit disk, and by \(C^k\) (respectively \(C_0^k\)) the \(k\)-dimensional polycylinder in which \(D_i=C\) (respectively \(C_0\)). Let \(\{a_{m,n}\}\) be a matrix of nonnegative numbers. By \(M(a_{m,n})\) we denote the space of all sequences of complex numbers \(x=\{t_n\}\) such that

\[ \|x\|_m=\sup_n |t_n|a_{m,n}<+\infty, \]

with convergence defined by the sequence of seminorms \(\|x\|_m\)*.

If in the nuclear space \(X\) there exists an unconditional basis \(e_n\)**, then this space is isomorphic to the space \(M(\|e_n\|_m)((^{10})\), Theorem 3). Theorems 1 and 2 are special cases of Theorem 3 \((^{10})\).

Theorem 1. \(H(C^k)\) is isomorphic to \(M(2^{m\sqrt[k]{n}})\).

Theorem 2. Let \(D\) be an arbitrary bounded complete circular domain*** in \(k\)-dimensional space. Then \(H(D)\) is isomorphic to \(M(2^{-\sqrt[k]{n/m}})\).

Theorem 3. Let \(D\) be a polycylinder, \(D=D_1\times D_2\times\cdots\times D_k\), such that: a) all domains \(D_i\) are finitely connected; b) for \(i=1,2,\ldots,p\) the complement of \(D_i\) consists only of isolated points; c) for \(i=p+1,p+2,\ldots,p+r\) the complement of \(D_i\) consists only of continua; d) for \(i=p+r+1,\ldots,k\) the complement of \(D_i\) contains both continua and isolated points.

Then the space \(H(D)\) is isomorphic to the space

\[ H(C^p\times C_0^{\,k-p})\times H(C^{p+1}\times C_0^{\,k-p-1})\times\cdots\times H(C^{k-r}\times C_0^r). \]

Theorem 4. Let \(D\) be a \(k\)-dimensional polycylinder. Then the space \(H(D)\) is isomorphic to some subspace of \(H(C_0^k)\). Moreover, if \(D\) is a bounded set, then \(H(C_0^k)\) is also isomorphic to some subspace of \(H(D)\). This means that the spaces \(H(D)\) and \(H(C_0^k)\) have one and the same linear dimension in the sense of Banach****.

Theorem 5. For complete circular polycylinders the relations

\[ d_a\bigl(H(C_0^k)\bigr)=d_a\bigl(H(C^{k-1}\times C_0)\bigr)=d_a\bigl(H(C^{k-2}\times C_0^2)\bigr)=\cdots \]

\[ \cdots=d_a\bigl(H(C\times C_0^{\,k-1})\bigr)<d_a\bigl(H(C_0^k)\bigr)<d_a\bigl(H(C^{k+1})\bigr). \]

hold.

Theorem 6. For any \(k\)-dimensional open set \(D\),

\[ d_a\bigl(H(C^k)\bigr)\leqslant d_a\bigl(H(D)\bigr)\leqslant d_a\bigl(H(C_0^k)\bigr). \]

* If these spaces are nuclear, then they are isomorphic to the corresponding Köthe spaces \((^7)\).

** The definition and properties of an unconditional basis are given in \((^2)\), pp. 73–77.

* A domain \(D\) is called a complete circular domain** with center \((a_1,\ldots,a_k)\) if, from the fact that \((z_1,\ldots,z_k)\in D\), it follows that
\((a_1+r(z_1-a_1)e^{i\varphi},\ldots,a_k+r(z_k-a_k)e^{i\varphi})\in D\) for all real \(\varphi\) and \(|r|\leqslant 1\) \((^3)\), p. 114.

**** For the definition of linear dimension in the sense of Banach, see \((^1)\), p. 193.

From Theorems 5 and 6 it follows:

Theorem 7. Let \(D\) be an open \(k\)-dimensional set, and \(D'\) an open \(k'\)-dimensional set. If \(k<k'\), then

\[ d_a\bigl(H(D)\bigr)<d_a\bigl(H(D')\bigr). \]

Received
16 IV 1960

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