MATHEMATICS

G. M. KESELMAN

ON A GENERALIZATION OF THE CONCEPT OF A BASIS AND OF N. K. BARI’S THEOREMS ON BASES

(Presented by Academician A. N. Kolmogorov, June 23, 1961)

In the present paper the concept of a continual basis in a Hilbert space is introduced and its properties are studied. It seems to us that the results obtained are a natural generalization of the results of N. K. Bari \((^{1,2})\), concerning discrete bases, and of the results of M. M. Dzhrbashyan and R. M. Martirosyan \((^3)\), concerning properties of biorthogonal kernels.

As is known, a sequence of elements \(\{g_n\}_{n=1}^{\infty}\) of a Hilbert space \(\mathfrak H\) is called its basis if for every \(x \in \mathfrak H\) there exists a unique numerical sequence \(\{\xi_n\}_{n=1}^{\infty}\) such that \(x=\sum_{n=1}^{\infty}\xi_n g_n\).

If for every \(x \in \mathfrak H\) the corresponding series \(\sum_{1}^{\infty}\xi_n g_n\) converges weakly to \(x\), then it also converges strongly (see \((^4)\), p. 310). Therefore a basis \(\{g_n\}_{n=1}^{\infty}\) may be identified with a linear mapping \(G\) of the space \(\mathfrak H\) into the space of numerical sequences: \((Gy)_n=(y,g_n)\), possessing the property that for every \(x \in \mathfrak H\) there exists a unique sequence \(\{\xi_n\}_{n=1}^{\infty}\) such that
\[ (x,y)=\sum_{1}^{\infty}\xi_n\,\overline{(G,y)_n} \]
for all \(y \in \mathfrak H\).

Let us introduce the following notation: \(\mathfrak H\) is a separable Hilbert space; \(\sigma\) is a nonnegative measure given on the Borel sets of the complex plane \(\Lambda\) (or, in general, \(\Lambda\) is a space with nonnegative measure \(\sigma\)); \(M_\sigma\) is the linear manifold of classes of \(\sigma\)-measurable and almost everywhere finite functions, equivalent with respect to the measure \(\sigma\).

Definition 1. A linear mapping \(G:\mathfrak H \to M_\sigma\) is called basic if for every element \(x \in \mathfrak H\) there exists a unique element \(\xi \in M_\sigma\) such that
\[ (x,y)=\int_{\Lambda}\xi(\lambda)\,\overline{(Gy)(\lambda)}\,\sigma(d\lambda) \quad \text{for all } y \in \mathfrak H . \tag{1} \]

Here and everywhere below, the integral \(\int_{\Lambda}\cdots\,\sigma(d\lambda)\) is understood as
\[ \lim_{r\to\infty}\int_{\Lambda_r\cap\Delta}\cdots\,\sigma(d\lambda), \]
where \(\Lambda_r=\{\lambda\in\Lambda:\ |\lambda|\le r\}\). This understanding of the integral also determines the manner of “ordering” the values of the mapping \((Gy)(\lambda)\). We note that in the case of a classical basis \(\{g_n\}_{n=1}^{\infty}\) we deal not simply with a countable set of elements, but with a set ordered in a definite way, i.e. with a sequence.

Putting \((Fx)(\lambda)=\xi(\lambda)\), we obtain a linear mapping \(F:\mathfrak H\to M_\sigma\), by means of which equality (1) can be given the form

\[ (x,y)=\int_\Lambda (Fx)(\lambda)\,\overline{(Gy)(\lambda)}\,\sigma(d\lambda). \tag{1'} \]

We shall call the mapping \(F\) conjugate with respect to \(G\).

Definition 2. The mapping \(G\) is called integrally bounded if there exists a \(\sigma\)-complete class \(\Omega\) of Borel sets \(\Delta\subset\Lambda\) such that

\[ \left|\int_\Delta (Gy)(\lambda)\,\sigma(d\lambda)\right|\leq K_\Delta\|y\|,\qquad K_\Delta<\infty,\ \Delta\in\Omega, \tag{2} \]

for all \(y\in\mathfrak H\). Here the class \(\Omega\) is called \(\sigma\)-complete if
\[ \int_\Delta \xi(\lambda)\,\sigma(d\lambda)=0 \]
for all \(\Delta\in\Omega\) only when \(\xi=0\)*.

If \(G\) is an integrally bounded basis mapping, then the conjugate mapping \(F\) is a basis mapping.

In view of (1′), it suffices to prove that
\[ \int_\Lambda \eta(\lambda)\,\overline{(Fx)(\lambda)}\,\sigma(d\lambda)=0 \]
for all \(x\in\mathfrak H\) only when \(\eta=0\). Let \(\Delta\) be an arbitrary fixed set in the class \(\Omega\). From inequality (2) we conclude that there exists an element \(x(\Delta)\in\mathfrak H\) such that

\[ (x(\Delta),y)=\int_\Delta \overline{(Gy)(\lambda)}\,\sigma(d\lambda),\qquad y\in\mathfrak H. \tag{3} \]

But the mapping \(G\) is a basis mapping; hence \(Fx(\Delta)=\chi_\Delta\), where \(\chi_\Delta\in M_\sigma\) is the class of functions equivalent to the characteristic function of the set \(\Delta\).

Now suppose that for some \(\eta\in M_\sigma\)
\[ \int_\Lambda \eta(\lambda)\,\overline{(Fx)(\lambda)}\,\sigma(d\lambda)=0 \]
for every \(x\in\mathfrak H\). Taking \(x=x(\Delta)\), \(\Delta\in\Omega\), we obtain
\[ \int_\Delta \eta(\lambda)\,\sigma(d\lambda)=0, \]
which is possible only for \(\eta=0\), as was required to prove.

Definition 3. A basis mapping \(G\) is called orthogonal if \(F=G\), where \(F\) is the conjugate mapping.

Every orthogonal mapping \(E\) is an isometric mapping of \(\mathfrak H\) onto \(L_\sigma^2\), and, conversely, every isometric mapping \(E\) of the space \(\mathfrak H\) onto \(L_\sigma^2\) is orthogonal.

From this the following corollary is easily obtained:

An orthogonal mapping is integrally bounded.

In what follows we shall consider only basis mappings \(G\) that are integrally bounded together with their conjugate mappings \(F\).

Definition 4. The mapping \(G\) is called a Bessel mapping if \(\mathfrak R(F)\subseteq L_\sigma^2\), i.e., for every \(x\in\mathfrak H\), \(Fx\in L_\sigma^2\) (\(\mathfrak R(F)\) is the range of the mapping \(F\)).

Theorem 1. In order that the mapping \(G\) be a Bessel mapping, it is necessary and sufficient that for every orthogonal mapping \(E\) there exist a linear bounded operator \(S:\mathfrak H\to\mathfrak H\) such that \(F=ES\). In this case \(S^{-1}\) exists and \(\mathfrak D(S^{-1})=\mathfrak R(S)=\mathfrak H\).

Sufficiency is obvious, since \(\mathfrak R(F)=\mathfrak R(ES)\subseteq\mathfrak R(E)=L_\sigma^2\).

* Definitions 1 and 2 were proposed by V. E. Lyantse.

Necessity. Since \(\mathfrak R(F)\subset L^2_\sigma\), putting

\[ s(x,y)=\int_\Lambda (Fx)(\lambda)\overline{(Ey)(\lambda)}\,\sigma(d\lambda), \tag{4} \]

for fixed \(x\) we have
\[ |s(x,y)|\le \|Fx\|_{L^2_\sigma}\|Ey\|_{L^2_\sigma} =\|Fx\|_{L^2_\sigma}\|y\|_{\mathfrak H}. \]
Consequently, \(\overline{s(x,y)}\) is a linear bounded functional in \(y\in\mathfrak H\): \(s(x,y)=(Sx,y)\), where \(S\) is a linear operator defined on all of \(\mathfrak H\).

Further,
\[ \left|s(x,E^{-1}\chi_\Delta)\right| = \left|\int_\Delta (Fx)(\lambda)\,\sigma(d\lambda)\right| \le K_\Delta\|x\|. \]
Thus, linear combinations of elements of the form \(E^{-1}\chi_\Delta\) belong to the domain of definition \(\mathfrak D(S^*)\) of the operator \(S^*\). Therefore \(\mathfrak D(S^*)\) is dense in \(\mathfrak H\), and since \(\mathfrak D(S)=\mathfrak H\), it follows that the operator \(S\) is bounded. Since \(E\) is an orthogonal mapping, we have
\[ s(x,y)=(Sx,y)=\int_\Lambda (ESx)(\lambda)\overline{(Ey)(\lambda)}\,\sigma(d\lambda). \]
Comparing this representation with (4), we arrive at the conclusion that \(F=ES\).

Let us prove that the operator \(S\) is invertible and that \(\mathfrak D(S^{-1})=\mathfrak R(S)=\mathfrak H\). Indeed, if \(Sx=0\), then \(Fx=0\) and \(x=0\), since the mapping \(F\) is basic. Consequently, \(S^{-1}\) exists. Next let \((Sx,y)=0\) for all \(x\). We shall prove that \(y=0\). We have
\[ 0=(Sx,y)=\int_\Lambda (Fx)(\lambda)\overline{(Ey)(\lambda)}\,\sigma(d\lambda). \]
Since the mapping \(F\) is basic, the last equality is possible only when \(Ey=0\), whence it follows that \(y=0\). The theorem is proved.

Definition 5. A mapping \(G\) is called Hilbertian if \(\mathfrak R(F)\supset L^2_\sigma\): for every \(\xi\in L^2_\sigma\) there exists an element \(x\in\mathfrak H\) such that \(Fx=\xi\).

Theorem 2. In order that the mapping \(G\) be Hilbertian, it is necessary and sufficient that for every orthogonal mapping \(E\) there exist a linear bounded operator \(T:\mathfrak H\to\mathfrak H\) such that \(FT=E\). In this case \(T^{-1}\) exists and \(\mathfrak D(T^{-1})=\mathfrak R(T)=\mathfrak H\).

Sufficiency is obvious, since
\[ \mathfrak R(F)\supset \mathfrak R(FT)=\mathfrak R(E)=L^2_\sigma. \]

Necessity. Let \(x\) be an arbitrary fixed element of the space \(\mathfrak H\). Since \(Ex\in L^2_\sigma\) and \(\mathfrak R(F)\supset L^2_\sigma\), the equation \(Fy=Ex\) is solvable with respect to \(y\in\mathfrak H\). It is easy to see that this solution is unique. Put \(y=Tx\). Then \(T\) is a linear operator defined on the whole space \(\mathfrak H\). Let us prove its boundedness. Let \(\Delta\) be an arbitrary bounded \(\sigma\)-measurable set. Then
\[ \left|(Tx,G^{-1}\chi_\Delta)\right| = \left|\int_\Delta (FTx)(\lambda)\,\sigma(d\lambda)\right| = \left|\int_\Delta (Ex)(\lambda)\,\sigma(d\lambda)\right| \le K_\Delta\|x\|, \]
since the mapping \(E\) is integrally bounded. From the inequality it follows that \(\mathfrak D(T^*)\) is dense in \(\mathfrak H\), and since \(\mathfrak D(T)=\mathfrak H\), the operator \(T\) is bounded. The invertibility of the operator \(T\) is obvious. Let us prove that \(\mathfrak R(T)=\mathfrak H\). Suppose for any \(x\in\mathfrak H\) that \((Tx,y)=0\). Then
\[ 0=(Tx,y)=\int_\Lambda (FTx)(\lambda)\overline{(Gy)(\lambda)}\sigma(d\lambda) = \int_\Lambda (Ex)(\lambda)\overline{(Gy)(\lambda)}\,\sigma(d\lambda). \]
It follows that \(Gy=0\), hence also \(y=0\), as was required to prove.

The following facts are easily established:

Theorem 3. If the mapping \(G\) is Besselian (Hilbertian), then the adjoint mapping \(F\) is Hilbertian (Besselian).

Corollary 1. In order that the mapping \(G\) be Besselian, it is necessary and sufficient that there exist a bounded positive operator \(A\) such that \(F=GA\).

Corollary 2. In order that the mapping \(G\) be Hilbertian, it is necessary and sufficient that there exist a bounded positive operator \(B\) such that \(G=FB\).

Definition 6. A mapping \(G\) is called a Fischer—Riesz mapping if it is simultaneously Besselian and Hilbertian.

From Theorem 3 and its corollaries there follows immediately

Theorem 4. In order that the mapping \(G\) be a Fischer—Riesz mapping, it is necessary and sufficient that there exist a bounded strictly positive operator \(C\) such that \(F=GC\).

Let us consider examples.

\(1^\circ\). Let \(g_1,g_2,\ldots\) be a basis in \(\mathfrak H\), and let \(\sigma\) be the measure concentrated at the points \(\lambda=1,2,\ldots,n,\ldots\), with \(\sigma(\{n\})=1\). Put \((Gy)(n)=(y,g_n)\). Then the mapping \(G\) is basis and integrally bounded. Indeed, the class \(\Omega\) of sets each of which consists of only one point (with a natural affix) is \(\sigma\)-complete, and

\[ \left|\int_{\Delta=\{n\}} (Gy)(\lambda)\,\sigma(d\lambda)\right| = |(y,g_n)|\leq \|g_n\|\,\|y\|. \]

Clearly, the adjoint mapping \(F\) is generated by the adjoint basis \(f_1,f_2,\ldots\).

In the example under consideration all properties of the mapping \(G\) coincide with the properties of the basis \(g_1,g_2,\ldots\), studied by N. K. Bari \((^1,^2)\).

\(2^\circ\). In the paper \((^3)\) M. M. Dzhrbashian and R. M. Martirosian considered “integral” operators \(G\) mapping the space \(L_{\sigma_1}^2(a_1,b_1)\) into \(L_{\sigma_2}^2(a_2,b_2)\), defined by relations of the form

\[ \int_{a_2}^{b_2} y(t)e_\xi(t)\,d\sigma_2(t) = \int_{a_1}^{b_1} x(t)\overline{G(t,\xi)}\,d\sigma_1(t),\qquad y=Gx, \]

where, for each fixed \(\xi\in(a_1,b_1)\), the kernel \(G(t,\xi)\in L_{\sigma_1}^2\); \(e_\xi(t)=1\), \(t\in[0,\xi)\), and \(e_\xi(t)=0\), \(t\notin[0,\xi)\), for \(\xi>0\); \(e_\xi(t)=-1\), \(t\in[\xi,0)\), and \(e_\xi(t)=0\), \(t\notin[\xi,0)\), for \(\xi<0\).

Identify the space \(\mathfrak H\) with \(L_{\sigma_1}^2\), and let \(\sigma=\sigma_2\). Then, as is not difficult to see, the mapping \(G\) generated by the kernel \(G(t,\xi)\) is integrally bounded. It is also easy to show that if the kernel \(G(t,\xi)\) is Besselian or Hilbertian (for the definitions see \((^3)\)), then the mapping \(G\) is Besselian or Hilbertian, respectively.

\(3^\circ\). Let \(\mathfrak H=L^2(-\infty,\infty)\), and let the measure \(\sigma\) be concentrated on the real axis and coincide there with Lebesgue measure. Define the mapping \(G\) by means of the equality

\[ (Gx)(\lambda)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{i\varphi(\lambda)t}x(t)\,dt, \]

where \(\varphi(\lambda)\) is a certain function.

Then

\[ (Fx)(\lambda)=\frac{\varphi'(\lambda)}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{i\varphi(\lambda)t}x(t)\,dt. \]

It is easy to arrive at the following conclusion: if \(\varphi(\lambda)\) increases monotonically, with \(\varphi(-\infty)=-\infty\), \(\varphi(\infty)=\infty\), then the mapping \(G\) is Besselian when

\[ \operatorname*{vrai\,max}_{-\infty<\lambda<\infty}\varphi'(\lambda)<\infty, \]

and Hilbertian when

\[ \operatorname*{vrai\,min}_{-\infty<\lambda<\infty}\varphi'(\lambda)>0. \]

Received
1 VI 1961

CITED LITERATURE

1. N. K. Bari, DAN, 54, 383 (1946).
2. N. K. Bari, Uch. zap. Mosk. univ., issue 148, 4, 69 (1951).
3. M. M. Dzhrbashian, R. M. Martirosian, DAN, 132, No. 5 (1960).
4. S. Kaczmarz, H. Steinhaus, Theory of Orthogonal Series, Moscow, 1958.