MATHEMATICS

Academician of the Academy of Sciences of the Armenian SSR M. M. DZHRBASHYAN

UNITARY PAIRS OF OPERATORS AND THEIR ANALYTIC CHARACTERISTIC IN THE SPACE \(L_2(a,b)\)

In the present note, as a natural generalization of the concept of a unitary operator, the concept of a unitary pair of operators acting in an abstract Hilbert space is introduced. Then an extension to unitary pairs of operators is given of the well-known theorem of S. Bochner \((^{1,3})\) on the analytic characteristic of unitary operators acting in the Hilbert space of functions \(L_2(a,b)\). We shall make essential use of this characteristic in our next communication, devoted to certain problems arising in connection with the well-known approximation theorem of Müntz \((^4)\).

\(1^\circ\). Let two linear bounded operators \(U_1\) and \(U_2\) be defined in an abstract Hilbert space \(H\), with ranges \(\Delta_k = U_k H \subseteq H\) \((k=1,2)\), and let \(U_1^*\) and \(U_2^*\) be the corresponding adjoint operators.

We shall call an ordered pair of operators \(\{U_1,U_2\}\) a unitary pair if, for arbitrary elements \(f\) and \(g\) of \(H\), the conditions

\[ (U_1^* f,\, U_1^* g) = (f,g); \tag{1} \]

\[ (U_1 f,\, U_1 g) + (U_2 f,\, U_2 g) = (f,g), \tag{2} \]

are satisfied, where \((f,g)\) is the scalar product generating the metric of the space \(H\).

We shall denote the class of all such unitary pairs by \(\mathfrak{A}(H)\). Let us note that if \(\{U_1,U_2\}\in \mathfrak{A}(H)\), then, generally speaking, \(\{U_2,U_1\}\notin \mathfrak{A}(H)\).

As is easy to see, conditions (1) and (2) are respectively equivalent to the following:

\[ U_1 U_1^* = I; \tag{1'} \]

\[ U_1^* U_1 + U_2^* U_2 = I, \tag{2'} \]

where \(I\) is the identity operator of the space \(H\).

The following assertions follow immediately from the definition of a unitary pair:

A. In order that a given linear bounded operator \(U_1\) in the space \(H\) be unitary, it is necessary and sufficient that from \(\{U_1,U_2\}\in \mathfrak{A}(H)\) it follow that \(U_2=O\) (where \(O\) is the operator carrying all elements of \(H\) into the zero element).

Indeed, the property of unitarity of the given operator \(U_1\) in \(H\) is characterized by the fulfillment of the two conditions

\[ U_1^* U_1 = U_1 U_1^* = I. \]

But if \(\{U_1,U_2\}\in \mathfrak{A}(H)\), then, in view of \((1')\) and \((2')\), these conditions are equivalent to the condition \(U_2^* U_2=0\), i.e. to the condition \(U_2=0\).

B. If \(\{U_1, U_2\}\in \mathfrak A(H)\), then the operators

\[ P_1=U_1^*U_1,\qquad P_2=U_2^*U_2 \tag{3} \]

project the space \(H\), respectively, onto the subspaces \(H_1\) and \(H_2\), which serve as orthogonal complements of one another.

Indeed, by \((1')\) and (3) we have

\[ P_1^2=P_1,\qquad P_1^*=P_1, \]

i.e. \(P_1\) is a projection operator mapping the space \(H\) onto some subspace \(H_1\). But, by \((2')\) and (3), \(P_2=I-P_1\), and therefore \(P_2\) is also a projection operator mapping the space \(H\) onto the subspace \(H_2=H\ominus H_1\).

Finally, as mutually orthogonal projection operators,

\[ P_1P_2=P_2P_1=0. \tag{4} \]

C. If \(\{U_1,U_2\}\in\mathfrak A(H)\), then for any elements \(f\) and \(g\) from \(\Delta_2=U_2H\) we also have

\[ (U_2^*f,U_2^*g)=(f,g). \tag{5} \]

Noting that \(I=P_1+P_2\) and taking into account (4), for arbitrary elements \(F\) and \(G\) from \(H\) we have:

\[ (F,G)=(P_1F+P_2F,\ P_1G+P_2G)=(P_1F,P_1G)+(P_2F,P_2G). \]

Hence, using the notation (3) and the conditions (1) and (2), we obtain

\[ (U_2^*U_2F,\ U_2^*U_2G)=(U_2F,\ U_2G), \]

and, since \(f=U_2F\subset\Delta_2,\ g=U_2G\subset\Delta_2\) are arbitrary, our assertion is proved.

D. If the linear bounded operator \(U_1\) satisfies condition (1), then there exists a linear bounded operator \(U_2\) (unique among positive and self-adjoint operators) such that \(\{U_1,U_2\}\in\mathfrak A(H)\).

Indeed, under condition \((1')\) the operators

\[ P_1=U_1^*U_1,\qquad P_2=I-P_1 \]

are mutually orthogonal projection operators. Consequently,

\[ P_2^2=P_2,\qquad P_2^*=P_2, \]

i.e. the operator \(P_2\) is self-adjoint and positive, since

\[ (P_2f,f)=(P_2^2f,f)=(P_2f,P_2f)\geq 0. \]

But then, by a known theorem \((^2)\), there exists a unique positive self-adjoint square root \(U_2=P_2^{1/2}\). The operator \(U_2\) will be the desired one.

Thus, if the operator \(U_1\) is not unitary, then, provided the operator \(U_1^*\) is isometric, it admits a “completion” by means of another operator \(U_2\), which together with it forms a unitary pair \(\{U_1,U_2\}\).

\(2^\circ.\) The linear system \(L_2(a,b)\) of functions measurable on the interval \((a,b)\) (where \(-\infty\leq a<b\leq+\infty\)) and with square-integrable modulus in the metric generated by the scalar product

\[ (f,g)=\int_a^b f(x)g(x)\,dx, \]

is a separable Hilbert space.

Define the function \(e_{\xi}(x)\), depending on the parameter \(\xi \ne 0\), as follows:

\[ e_{\xi}(x)= \begin{cases} 1, & x\in[0,\xi),\\ 0, & x\notin[0,\xi), \end{cases} \quad \xi>0; \qquad e_{\xi}(x)= \begin{cases} -1, & x\in[\xi,0),\\ 0, & x\notin[\xi,0). \end{cases} \quad \xi<0, \tag{6} \]

and, for simplicity, assume that \(x=0\) is an interior or boundary point of the interval \((a,b)\).

Unitary pairs of operators acting in the space \(L_2(a,b)\) can be characterized analytically by means of the following theorem:

Theorem 1. To any unitary pair of operators \(\{U_1,U_2\}\in \mathfrak A(H)\), acting in the space \(H=L_2(a,b)\), there correspond four functions

\[ K(\xi,x)=U_1 e_{\xi}(x),\qquad K^{*}(\xi,x)=U_1^{*}e_{\xi}(x), \]

\[ R(\xi,x)=U_2 e_{\xi}(x),\qquad R^{*}(\xi,x)=U_2^{*}e_{\xi}(x), \tag{7} \]

belonging to this space for each fixed \(\xi\in(a,b)\) and possessing the property that the correspondence

\[ g_1=U_1 f,\qquad g_2=U_2 f,\qquad f=U_1^{*}g_1+U_2^{*}g_2,\qquad f\in H, \tag{8} \]

is realized by means of the formulas

\[ \int_{0}^{\xi} g_1(x)\,dx = \int_{a}^{b} \overline{K^{*}(\xi,x)}\,f(x)\,dx, \qquad \int_{0}^{\xi} g_2(x)\,dx = \int_{a}^{b} \overline{R^{*}(\xi,x)}\,f(x)\,dx, \tag{9} \]

\[ \int_{0}^{\xi} f(x)\,dx = \int_{a}^{b} \overline{K(\xi,x)}\,g_1(x)\,dx + \int_{a}^{b} \overline{R(\xi,x)}\,g_2(x)\,dx, \tag{10} \]

while the correspondence

\[ g=U_1^{*}f,\qquad f=U_1g,\qquad f\in H, \tag{11} \]

is realized by means of the formulas

\[ \int_{0}^{\xi} g(x)\,dx = \int_{a}^{b} \overline{K(\xi,x)}\,f(x)\,dx, \qquad \int_{0}^{\xi} f(x)\,dx = \int_{a}^{b} \overline{K^{*}(\xi,x)}\,g(x)\,dx. \tag{12} \]

Moreover, the functions (7) satisfy the equations:

\[ \begin{aligned} \text{a)}\quad& \int_{a}^{b} \overline{K(\xi,x)}\,K(\eta,x)\,dx + \int_{a}^{b} \overline{R(\xi,x)}\,R(\eta,x)\,dx \\ \text{b)}\quad& \int_{a}^{b} \overline{K^{*}(\xi,x)}\,K^{*}(\eta,x)\,dx \end{aligned} \left\} = \int_{a}^{b} e_{\xi}(x)e_{\eta}(x)\,dx = \begin{cases} \min(|\xi|,|\eta|), & \text{when } \xi\eta\ge 0,\\ 0, & \text{when } \xi\eta\le 0; \end{cases} \]

\[ \text{c)}\quad \int_{0}^{\eta} K(\xi,x)\,dx = \int_{0}^{\xi} \overline{K^{*}(\eta,x)}\,dx; \]

\[ \text{d)}\quad \int_{0}^{\eta} R(\xi,x)\,dx = \int_{0}^{\xi} \overline{R^{*}(\eta,x)}\,dx. \]

Conversely, every quadruple of functions

\[ K(\xi,x),\ K^*(\xi,x);\ R(\xi,x),\ R^*(\xi,x), \]

satisfying equations a), b), c), d), generates, according to formulas (9)—(10), (12), a certain unitary pair of operators \(\{U_1,U_2\}\), connected with these functions by formulas (7).

Let us note that, in view of assertion A, Theorem 1 contains Bochner’s theorem on the characteristic of unitary operators in \(L_2(a,b)\). In view of this, each of the functions \(K(\xi,x)\) and \(K^*(\xi,x)\) of Theorem 1 is naturally called the kernel of a unitary operator, if \(U_2=0\), i.e. if

\[ R(\xi,x)\equiv R^*(\xi,x)\equiv 0. \]

It is known\(^3\) that if the function \(K^*(\xi,x)\) satisfies equation b) and, moreover, is complete, i.e. from

\[ \int_a^b K^*(\xi,x)f(x)\,dx=0,\qquad \xi\in(a,b), \]

where \(f(x)\in L_2(a,b)\), it follows that \(f(x)\equiv 0\), then \(K^*(\xi,x)\) is the kernel of some unitary operator.

When the completeness condition of the kernel \(K^*(\xi,x)\) is removed from assertion Г, in view of Theorem 1 it follows:

Theorem 2. Let the function \(K^*(\xi,x)\) satisfy equation b), i.e.

\[ \int_a^b \overline{K^*(\xi,x)}\,K^*(\eta,x)\,dx = \int_a^b e_\xi(x)e_\eta(x)\,dx;\qquad \xi,\eta\in(a,b). \]

Then there exists a unique function \(K(\xi,x)\) and functions \(R(\xi,x)\), \(R^*(\xi,x)\), which, together with \(K^*(\xi,x)\), also satisfy equations a), c), d) of Theorem 1, thereby generating a certain unitary pair of operators \(\{U_1,U_2\}\) in \(L_2(a,b)\) according to formulas (9)—(10), (12).

The functions \(R(\xi,x)\), \(R^*(\xi,x)\) are also unique if one requires that

\[ R^*(\xi,x)\equiv R(\xi,x). \]

This theorem may be regarded as a fundamental solution of the question of completing incomplete kernels. Realizations of such a completion in some examples important for analysis will be given by us later.

Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR

Received
21 VIII 1961

REFERENCES

\(^1\) S. Bochner, K. Chandrasekharan, Fourier Transforms, Princeton, 1949, pp. 150—156.
\(^2\) F. Riesz, B. Sz.-Nagy, Lectures on Functional Analysis, Moscow, 1954, pp. 316—317.
\(^3\) M. M. Dzhrbashyan, R. M. Martirosyan, DAN, 132, No. 5 (1960).
\(^4\) M. M. Dzhrbashyan, DAN, 141, No. 3 (1961).