Reports of the Academy of Sciences of the USSR

1. Volume 144, No. 3

MATHEMATICS

A. V. STRAUS

ON SELF-ADJOINT OPERATORS IN THE ORTHOGONAL SUM OF HILBERT SPACES

(Presented by Academician A. N. Kolmogorov 12 I 1962)

Recall that a linear operator (A), acting in a Hilbert space (H), is called Hermitian if for any elements (f, g) from its domain of definition (D_A), ((Af, g) = (f, Ag)). If, in addition, the manifold (D_A) is dense in (H), then the operator (A) is called symmetric. M. A. Naimark ((^{1,2})) described the structure of a self-adjoint operator that acts in the orthogonal sum of two Hilbert spaces and is an extension of a closed symmetric operator acting in one of the spaces. This description is given in terms connected with the Cayley transform. If, instead of a symmetric operator in one of the spaces, a closed Hermitian operator is given, then the description of all possible self-adjoint extensions of such an operator, also based on the Cayley transform, is given by results of M. A. Krasnoselskii ((^{3,4})). In the present paper we study the structure of a self-adjoint operator in the orthogonal sum of two Hilbert spaces under the assumption that it is an extension of a closed Hermitian operator with finite defect numbers, acting in one of the spaces. The structure of such a self-adjoint operator is described in terms of the so-called boundary spaces and boundary operators.

1. A linear operator (A), acting in the orthogonal sum (H = H_1 \oplus H_2) of Hilbert spaces (H_1) and (H_2), will be called a simple coupling of linear operators (B_1, B_2), acting respectively in (H_1) and (H_2), if (P_k D_A = D_{B_k}) and for any (f \in D_A), (P_k Af = B_k P_k f) ((k = 1, 2)), where (P_k) is the projection operator in (H) onto (H_k).

For the study of simple couplings that are Hermitian operators, we shall use the concepts of a boundary space and a boundary operator introduced in ((^5)). Recall that a linear space (\mathcal L) with a nondegenerate scalar product ([\varphi,\psi]), in general indefinite, is called a boundary space of a linear operator (B), acting in a Hilbert space, if there exists a linear operator (\Gamma), mapping (D_B) onto (\mathcal L), such that for any

[
f, g \in D_B \qquad [\Gamma f, \Gamma g] = \frac{1}{i}\bigl[(Bf, g) - (f, Bg)\bigr];
]

the operator (\Gamma) is then called a boundary operator.

Lemma. Let the linear operators (B_1) in (H_1) and (-B_2) in (H_2) have a common boundary space (\mathcal L), and let (\Gamma_{B_1}, \Gamma_{-B_2}) be some of their boundary operators with common range (\mathcal L). Then the simple coupling (A) of the operators (B_1, B_2), defined on the manifold (D_A) of all possible elements ({f_1, f_2} \in H_1 \oplus H_2) such that (f_1 \in D_{B_1}), (f_2 \in D_{B_2}), (\Gamma_{B_1} f_1 = \Gamma_{-B_2} f_2), is a Hermitian operator. Conversely, if the simple coupling (A) of the operators (B_1, B_2) is a Hermitian operator, then (B_1, -B_2) have a common boundary space, and the domain of definition (D_A) of the operator (A) is specified in the manner indicated above by means of certain boundary operators (\Gamma_{B_1}, \Gamma_{-B_2}).

We shall agree to say that in the case described in the lemma, the simple coupling (A) of the operators (B_1, B_2) is defined by the boundary operators (\Gamma_{B_1}, \Gamma_{-B_2}).

If (A) is any linear operator in a Hilbert space (H), and (H_1) is a subspace in (H), then by a part of the operator (A) generated in (H_1) we shall mean the operator (A_1 \subset A), defined on the manifold (D_{A_1}) of all (f \in D_A \cap H_1) for which (Af \in H_1). In the case when (A) is a closed operator, (A_1) is also closed. If (A) is a self-adjoint operator in (H = H_1 \oplus H_2), then its parts (A_1) and (A_2), generated respectively in (H_1) and (H_2), are closed Hermitian operators, and the defect indices of the operators (A_1) and (-A_2) are the same.

Theorem 1. Let (A_1, A_2) be closed symmetric operators in (H_1, H_2) such that the defect numbers of the operators (A_1, -A_2), respectively, are equal and finite. Then the operators (A_1^, -A_2^) possess a common boundary space (u), whatever the boundary operators (\Gamma_{A_1^}, \Gamma_{-A_2^}) with common range defining it may be; the simple coupling (A) of the operators (A_1^, A_2^) is a self-adjoint operator in (H = H_1 \oplus H_2). Moreover, the parts of the operator (A) generated in (H_1, H_2) coincide respectively with (A_1, A_2).

Theorem 2. Let (A) be a self-adjoint operator in (H = H_1 \oplus H_2). If its parts (A_1, A_2), generated respectively in (H_1, H_2), are symmetric operators with finite defect numbers, then the operator (A) is a simple coupling of the operators (A_1^, A_2^), defined by some boundary operators (\Gamma_{A_1^}, \Gamma_{-A_2^}).

2. Let (A) be a closed Hermitian operator in (H). A closed linear operator (B) in (H), which is an extension of the operator (A) and has domain (D_B) dense in (H), will be called a special extension of the operator (A) if the linear manifold (G_B) of all elements (f \in D_B \cap D_{B^}) for which (Bf = B^f) coincides with (D_A). From this definition it follows that, simultaneously with (B), (B^*) is also a special extension of the operator (A).

We note the particular case when the operator (A) is symmetric. In order that a closed operator (B) be a special extension of a closed symmetric operator (A), it is necessary and sufficient that the conditions hold: 1) (A \subset B \subset B^); 2) (D_B \cap D_{B^} = D_A). In particular, (A) and (A^*) are special extensions of the operator (A).

Suppose that the closed Hermitian operators (A_1) and (-A_2), acting respectively in (H_1) and (H_2), have the same defect indices. Let (B_1) in (H_1) and (B_2) in (H_2) be special extensions of the operators (A_1) and (A_2) such that the operators (B_1) and (-B_2) have a common boundary space (\mathcal L), while (-B_1^) and (B_2^) have a common boundary space (\mathcal L'). Consider any simple couplings (C) and (C') of the operators (B_1, B_2) and (B_1^, B_2^), defined respectively by the boundary operators (\Gamma_{B_1}, \Gamma_{-B_2}) and (\Gamma_{-B_1^}, \Gamma_{B_2^}). The operators (C) and (C') coincide on the manifold (D_C \cap D_{C'}) if and only if (D_C \cap D_{C'} = D_{A_1} \oplus D_{A_2}), and this is equivalent to the following condition imposed on the boundary operators: if, for some elements (f_1 \in D_{B_1} \cap D_{B_1^}) and (f_2 \in D_{B_2} \cap D_{B_2^}),
(\Gamma_{B_1} f_1 = \Gamma_{-B_2} f_2) and (\Gamma_{-B_1^} f_1 = \Gamma_{B_2^} f_2), then (f_1 \in D_{A_1}) and (f_2 \in D_{A_2}). Assuming this condition to be fulfilled,* we shall call the minimal common linear extension (A)

* In that particular case when the defect numbers of the operators (A_1) and (A_2) are finite, it is enough to assume that the common boundary space (\mathcal L) is possessed by the operators (B_1) and (B_2); hence it already follows that the operators (-B_1^) and (B_2^) also possess a common boundary space (\mathcal L').

* If at least one of the operators (A_1) or (A_2) is symmetric, then the indicated condition is fulfilled for any boundary operators (\Gamma_{B_1}, \Gamma_{-B_2}, \Gamma_{-B_1^}, \Gamma_{B_2^*}).

operators (C) and (C'), by the double coupling of the operators (B_1, B_2, B_1^, B_2^), defined by the boundary operators (\Gamma_{B_1}, \Gamma_{-B_2}, \Gamma_{-B_1^}, \Gamma_{B_2^}); the operators (B_1, B_2, B_1^, B_2^) shall conditionally be called the components of the double coupling (A).

Theorem 3. Let the operators (B_1) in (H_1) and (B_2) in (H_2) be special extensions of the closed Hermitian operators (A_1, A_2) such that (B_1, -B_2), and also (-B_1^, B_2^), have common boundary spaces. Then the double coupling (A) of the operators (B_1, B_2, B_1^, B_2^), defined by the boundary operators (\Gamma_{B_1}, \Gamma_{-B_2}, \Gamma_{-B_1^}, \Gamma_{B_2^}), is a Hermitian operator in (H=H_1\oplus H_2), and its parts generated in (H_1, H_2) coincide respectively with the operators (A_1, A_2). If the deficiency numbers of the operators (A_1, A_2) are finite, then (A) is a self-adjoint operator.

Theorem 4. Let (A) be a self-adjoint operator in (H=H_1\oplus H_2) such that its parts (A_1, A_2), generated respectively in (H_1, H_2), have finite deficiency numbers. Then there exist special extensions (B_1, B_2) of the operators (A_1, A_2), respectively, such that (A) is the double coupling of the operators (B_1, B_2, B_1^, B_2^), defined by certain boundary operators (\Gamma_{B_1}, \Gamma_{-B_2}, \Gamma_{-B_1^}, \Gamma_{B_2^}).

We note that, under the conditions of Theorem 4, the operators (B_1) in (H_1) and (B_2) in (H_2), which are components of the double coupling equal to the operator (A), are in general not uniquely determined by this operator. However, as is easily seen, from the operators (A) and (B_1) the operator (B_2) is reconstructed uniquely; its domain (D_{B_2}) is the manifold of all elements (f_2\in H_2) for each of which there exists an element (f_1\in D_{B_1}) such that (f_1+f_2\in D_A) and (P_1A(f_1+f_2)=B_1f_1); in this case (B_2f_2=P_2A(f_1+f_2)); (P_1) and (P_2) denote, as before, the projection operators in (H) onto (H_1) and (H_2), respectively.

Theorem 5. Let the operators (A) in (H=H_1\oplus H_2), (A_1) in (H_1), and (A_2) in (H_2) be the same as in Theorem 4. A special extension (B_1) of the operator (A_1) is a component of some double coupling equal to the operator (A) if and only if the following conditions are satisfied:

1) (B_1f_1=P_1Af_1) only when (f_1\in D_{A_1}); 2) (B_1^*g_1=P_1Ag_1) only when (g_1\in D_{A_1}).

If the operator (A_2) is symmetric, then every special extension (B_1) of the operator (A_1) satisfies these conditions*.

1. Let us illustrate the preceding constructions on the simplest examples.

1) Let (H=\mathcal L^2(0,2)), (H_1=\mathcal L_2(0,1)), (H_2=\mathcal L^2(1,2)); let (A) be the differentiation operator in (\mathcal L^2(0,2)) generated by the operation (i\dfrac{d}{dx}) and the boundary condition (y(0)=y(2)). (A_1) and (A_2) are symmetric differentiation operators in (\mathcal L^2(0,1)) and (\mathcal L^2(1,2)), defined respectively by the boundary conditions:
(y(0)=y(1)=0,\quad y(1)=y(2)=0.)

A common boundary space of the operators (A_1^) and (-A_2^) is a two-dimensional space with an indefinite metric. The operator (A) is the simple coupling of the operators (A_1^) and (A_2^), defined by the boundary operators (\Gamma_{A_1^}) and (\Gamma_{-A_2^}), which are given by the formulas:

[
\Gamma_{A_1^}f_1={f_1(1),\, f_1(0)}\quad (f_1\in D_{A_1^});\qquad
\Gamma_{-A_2^}f_2={f_2(1),\, f_2(2)}\quad (f_2\in D_{A_2^}).
]

2) Let (H, H_1, H_2) and the operator (A) be the same as in 1). Consequently, the operators (A_1, A_2) also remain the same. Let (B_1) be an operator, ge-

* We note that among the special extensions (B_1) of the operator (A_1) satisfying the conditions of Theorem 5 there are some that possess regular points in the complex plane. This fact is essential for applications of the results presented to the theory of characteristic functions of linear operators, on which we do not dwell here.

generated in (\mathcal L^2(0,1)) by the operation (i\dfrac{d}{dx}) and the boundary condition (y(0)=0). (B_1) is a special extension of the operator (A_1). The corresponding special extension (B_2) of the operator (A_2) is generated in the space (\mathcal L^2(1,2)) by the same operation (i\dfrac{d}{dx}) and the boundary condition (y(2)=0). The boundary space of each of the operators (B_1, -B_2, -B_1^, B_2^) is a one-dimensional unitary space. The operator (A) is a double coupling of the operators (B_1, B_2, B_1^, B_2^), defined by the boundary operators (\Gamma_{B_1}, \Gamma_{-B_2}, \Gamma_{-B_1^}, \Gamma_{B_2^}), which are given by the formulas:

[
\Gamma_{B_1} f_1=f_1(1)\quad (f_1\in D_{B_1});\qquad
\Gamma_{-B_2} f_2=f_2(1)\quad (f_2\in D_{B_2});
]

[
\Gamma_{-B_1^} g_1=g_1(0)\quad (g_1\in D_{B_1^});\qquad
\Gamma_{B_2^} g_j=g_2(2)\quad (g_2\in D_{B_2^}).
]

3) Let the space (H) be countably dimensional and let (e_0,e_1,e_2,\ldots,e_k,\ldots) be an orthonormal basis in (H). Consider the linear symmetric operator (A') in (H), defined on all possible finite linear combinations of the elements of the basis according to the formulas:

[
A'e_0=a_0e_0+b_0e_1,\qquad
A'e_k=b_{k-1}e_{k-1}+a_ke_k+b_ke_{k+1}\quad (k=1,2,\ldots),
]

where (a_k) and (b_k) are real numbers, with (b_k\ne0) ((k=0,1,2,\ldots)). Denote the closure of the operator (A') by (A). As is known, the operator (A) is either self-adjoint or symmetric with defect index ((1.1)). Suppose that the first case occurs. Fixing some natural number (m), denote by (H_1) the linear span of the system of elements (e_0,e_1,\ldots,e_m), and put (H_2=H\ominus H_1). It is easy to see that the parts (A_1) and (A_2) of the operator (A), generated respectively in (H_1) and (H_2), are Hermitian operators with defect index ((1.1)) and have nondense domains of definition in these subspaces. Define the operator (B_1) in (H_1) as a linear extension of the operator (A_1), setting

[
B_1e_m=b_{m-1}e_{m-1}+(a_m+ib_m)e_m.
]

The operator (B_1) is a special extension of the operator (A_1), satisfying all the conditions of Theorem 5. Therefore (A) can be represented in the form of a double coupling, one of whose components is (B_1). The second component (B_2) is now determined uniquely: (B_2) is the linear extension of the operator (A_2), defined on the linear span of the manifold (D_{A_2}) and the element (e_{m+1}) by the formula

[
B_2e_{m+1}=(a_{m+1}-ib_m)e_{m+1}+b_{m+1}e_{m+2}.
]

The operators (B_1,-B_2,-B_1^,B_2^) have a common one-dimensional boundary space. Define the boundary operators as follows:

[
\Gamma_{B_1}f_1=\Gamma_{-B_1^*}f_1=(f_1,e_m)\quad (f_1\in H_1);
]

[
i\Gamma_{-B_2}f_2=-i\Gamma_{B_2^}f_2=(f_2,e_{m+1})\quad (f_2\in D_{B_2}=D_{B_2^}).
]

It is easy to verify that (A) is a double coupling of the operators (B_1,B_2,B_1^,B_2^), defined by the indicated boundary operators.

Received
19 XII 1961

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