Abstract
Full Text
Reports of the Academy of Sciences of the USSR
- Volume 132, No. 5
MATHEMATICS
Academician of the Academy of Sciences of the Armenian SSR M. M. Dzhrbashyan and R. M. Martirosyan
On the General Theory of Biorthogonal Kernels
In a note by one of the authors ((^1)), questions concerning quasi-isometric mappings of Hilbert spaces of functions (L_\sigma^2) were studied and, in particular, an analytic characterization of such mappings was given, generalizing the classical Bochner theorem ((^2)). The present note is essentially a continuation of that investigation. Here the notions of Bessel and Hilbert kernels are introduced and some of their properties are established, which are natural continual analogues of results of N. K. Bari ((^{3-5})).
Let (\sigma_k(x)) ((k=1,2)) be a nondecreasing function, defined and continuous from the right on the interval ((a_k,b_k)), where (-\infty \leq a_k < b_k \leq +\infty), and of bounded variation on every segment ([\alpha,\beta]\subset(a_k,b_k)). Denote by (H_k=L_{\sigma_k}^2(a_k,b_k)) the Hilbert space of all (\sigma_k)-measurable functions on ((a_k,b_k)) that are square-summable.
As shown in ((^1)), if the operators (R_1) and (R_{*1}), mapping all of (H_1) onto all of (H_2), form an isometric pair, i.e.
[
(R_1 f_1,R_{1} f_2){\sigma_2}=(R{1} f_1,R_1 f_2){\sigma_2}=(f_1,f_2)\quad (f_1,f_2\in H_1),
]
where
[
(f,g){\sigma_k}=\int\,d\sigma_k(x)}^{b_k} f(x)\overline{g(x)
]
is the scalar product in the corresponding space (H_k), then there exist kernels (\widetilde K(\xi,x)) and (\widetilde K_*(\xi,x)) such that
[
\widetilde K(\xi,x)\in H_1,\qquad \widetilde K_*(\xi,x)\in H_1\quad (\xi\in(a_2,b_2))
\tag{1}
]
and for every (f(x)\in H_1) we have
[
\int_{a_2}^{b_2} g(x)e_\xi(x)\,d\sigma_2(x)
=
\int_{a_1}^{b_1} \overline{\widetilde K(\xi,x)}\,f(x)\,d\sigma_1(x),
\tag{2}
]
[
\int_{a_2}^{b_2} g^(x)e_\xi(x)\,d\sigma_2(x)
=
\int_{a_1}^{b_1} \overline{\widetilde K_(\xi,x)}\,f(x)\,d\sigma_1(x),
\tag{3}
]
where (g(x)=R_1 f(x)) and (g^(x)=R_{1}f(x)). Here
[
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{4}
]
We note that (\widetilde K(\xi,x)) and (\widetilde K_*(\xi,x)) are related by
[
\int_{a_1}^{b_1} \overline{\widetilde K(\xi,x)}\,\widetilde K_*(\eta,x)\,d\sigma_1(x)
=
\int_{a_2}^{b_2} e_\xi(x)e_\eta(x)\,d\sigma_2(x).
\tag{5}
]
and both are complete. In this connection we call a certain kernel (K(\xi,x)) complete in (H_1), if from the fact that (f(x)\in H_1) and
[
\int_{a_1}^{b_1} K(\xi,x)\overline{f(x)}\,d\sigma_1(x)=0
]
for all (\xi\in(a_2,b_2)), it follows that (f(x)\equiv 0). In the particular case of an isometric operator (R), mapping all of (H_1) onto all of (H_2), we have (R=R_1=R_{1}). In this case (\widetilde K(\xi,x)\equiv \widetilde K_(\xi,x)\equiv K(\xi,x)), and formulas (2), (3), and (5) hold if everywhere (K(\xi,x)) and (\widetilde K_(\xi,x)) are replaced by (K(\xi,x)), and one sets (g(x)=g^(x)=Rf(x)).
It is useful to note the following theorem.
Theorem 1. Suppose that for all (\xi\in(a_2,b_2)) we have (K(\xi,x)\in H_1), where (K(\xi,x)) is complete in (H_1) and satisfies the condition
[
\int_{a_1}^{b_1}\overline{K(\xi,x)}K(\eta,x)\,d\sigma_1(x)
=
\int_{a_2}^{b_2} e_\xi(x)e_\eta(x)\,d\sigma_2(x),
\qquad
\xi,\eta\in(a_2,b_2),
\tag{6}
]
for all (\xi,\eta\in(a_2,b_2)). Then there exist a kernel (H(\xi,x)) complete in (H_2) and an isometric operator (V), mapping all of (H_2) onto all of (H_1), such that if one sets (g(x)=V^{-1}f(x)), then for all (f(x)\in H_1) and all (\xi\in(a_2,b_2))
[
\int_{a_2}^{b_2} g(x)e_\xi(x)\,d\sigma_2(x)
=
\int_{a_1}^{b_1}\overline{K(\xi,x)}f(x)\,d\sigma_1(x),
]
[
\int_{a_1}^{b_1} f(x)e_\xi(x)\,d\sigma_1(x)
=
\int_{a_2}^{b_2}\overline{H(\xi,x)}g(x)\,d\sigma_2(x),
]
and the relations
[
\int_{a_2}^{b_2}\overline{H(\xi,x)}H(\eta,x)\,d\sigma_2(x)
=
\int_{a_1}^{b_1} e_\xi(x)e_\eta(x)\,d\sigma_1(x),
\qquad
\xi,\eta\in(a_1,b_1),
]
[
\int_{a_1}^{b_1} K(\xi,x)e_\eta(x)\,d\sigma_1(x)
=
\int_{a_2}^{b_2}\overline{H(\eta,x)}e_\eta(x)\,d\sigma_2(x),
\qquad
\xi\in(a_2,b_2),\quad \eta\in(a_1,b_1).
]
This theorem justifies the following definition. Every complete kernel (K(\xi,x)) is called the kernel of an isometric operator if, for all (\xi,\eta\in(a_2,b_2)), condition (6) is satisfied.
We shall further say that the function (\widetilde K(\xi,x)), defined for all (\xi\in(a_2,b_2)) and (x\in(a_1,b_1)), is a (B)-kernel if there exists a function (\widetilde K_(\xi,x)), defined for the same (\xi) and (x), such that (\widetilde K(\xi,x)) and (\widetilde K_(\xi,x)) satisfy conditions (1) and (5) and, moreover, both are complete. In this connection we shall call (\widetilde K_(\xi,x)) the kernel conjugate to* (\widetilde K(\xi,x)). It is clear that the conjugate kernel is determined uniquely. It is also obvious that a kernel conjugate to a (B)-kernel is itself a (B)-kernel.
We shall further say that a (B)-kernel (\widetilde K(\xi,x)) is nonless, if to every function (f(x)\in H_1) there corresponds a certain function (g(\xi)\in H_2) such that for all (\eta\in(a_2,b_2))
[
\int_{a_1}^{b_1} f(x)\overline{\widetilde K_*(\eta,x)}\,d\sigma_1(x)
=
\int_{a_2}^{b_2} g(\xi)e_\eta(\xi)\,d\sigma_2(\xi).
\tag{7}
]
Obviously, (g(\xi)) is determined uniquely.
Theorem 2. In order that a (B)-kernel (\widetilde K(\xi,x)) be nonless, it is necessary and sufficient that there exist such a linear bounded
operator (A), mapping (H_1) into itself, such that for all (\xi \in (a_2,b_2))
[
A\widetilde K(\xi,x)=K(\xi,x),
\tag{8}
]
where (K(\xi,x)) is the kernel of some isometric operator mapping (H_1) onto (H_2).
We now introduce the notion of a Hilbert kernel. We shall say that the (B)-kernel (\widetilde K_*(\xi,x)) is a Hilbert kernel if to every function (g(\xi)\in H_2) there corresponds some function (f(x)\in H_1) such that
[
\int_{a_1}^{b_1} f(x)\,\overline{\widetilde K(\eta,x)}\,d\sigma_1(x)
=
\int_{a_2}^{b_2} g(\xi)e_\eta(\xi)\,d\sigma_2(\xi).
\tag{9}
]
It is then obvious that the function (f(x)) is determined uniquely.
Theorem 3. In order that the (B)-kernel (\widetilde K_*(\xi,x)) be a Hilbert kernel, it is necessary and sufficient that there exist a linear bounded operator (C), mapping (H_1) into itself, such that for all (\xi\in(a_2,b_2))
[
\widetilde K_*(\xi,x)=C K(\xi,x),
\tag{10}
]
where (K(\xi,x)) is the kernel of some isometric operator.
Theorem 4. If either of the conditions (8) or (10) is satisfied, the corresponding operators (A) or (C) are invertible in the sense that from (Af=0) or (Cf=0) it follows that (f=0).
Theorem 5. If a certain (B)-kernel is Bessel, then the kernel adjoint to it is Hilbert, and conversely.
Theorem 6. In order that the (B)-kernel (\widetilde K(\xi,x)) be Bessel, it is necessary and sufficient that there exist a positive bounded Hermitian operator (T), defined on (H_1), such that for all (\xi\in(a_2,b_2)) the condition
[
\widetilde K_*(\xi,x)=T\widetilde K(\xi,x)
\tag{11}
]
be satisfied.
It can be proved that, when condition (11) is satisfied, the operator is invertible, i.e., from (Tf=0) it follows that (f=0).
From this theorem there immediately follows the analogous condition for Hilbert kernels.
We shall now agree to call a certain (B)-kernel a Riesz–Fischer kernel if it is simultaneously Bessel and Hilbert. It is easy to see that if one of two adjoint (B)-kernels is a Riesz–Fischer kernel, then the other is as well.
Theorem 7. If (\widetilde K(\xi,x)) is a Riesz–Fischer kernel, then for every (f(x)\in H_1) there exist uniquely determined functions (g(x)\in H_2) and (g^*(x)\in H_2) such that the equalities
[
\int_{a_2}^{b_2} g(x)e_\xi(x)\,d\sigma_2(x)
=
\int_{a_1}^{b_1} \overline{\widetilde K(\xi,x)}\,f(x)\,d\sigma_1(x);
]
[
\int_{a_2}^{b_2} g^(x)e_\xi(x)\,d\sigma_2(x)
=
\int_{a_1}^{b_1} \overline{\widetilde K_(\xi,x)}\,f(x)\,d\sigma_1(x).
]
At the same time there exist positive constants (M,m,K,k) such that the inequalities
[
m|f|{\sigma_1}\leqslant |g|\leqslant M|f|{\sigma_1};
\qquad
k|f|\leqslant |g^*|{\sigma_2}\leqslant K|f|.
]
hold.
Theorem 8. In order that the (B)-kernel (\widetilde K(\xi,x)) be a Riesz–Fischer kernel, it is necessary and sufficient that there exist a linear bounded operator (A), mapping (H_1) onto all of (H_1), such that (A\widetilde K(\xi,x)=)
[
=K(\xi,x),
]
where (K(\xi,x)) is the kernel of some isometric operator. In this case the operator (A) has a bounded inverse.
Theorem 9. In order that the (B)-kernel (\widetilde K(\xi,x)) be a Riesz—Fischer kernel, it is necessary and sufficient that there exist a bounded, positive-definite Hermitian operator (T), defined on (H_1), such that for all (\xi\in(a_2,b_2)) condition (11) be satisfied.
We shall now adopt the following definition. A (B)-kernel (\widetilde K(\xi,x)) is called a basis kernel or a Riesz kernel in (H_1) if there exists a kernel (\widetilde H_*(\xi,x)), complete in (H_2) ((x\in(a_2,b_2))), connected with (\widetilde K(\xi,x)) by the relation
[
\int_{a_1}^{b_1}\widetilde K(\eta,x)e_\xi(x)\,d\sigma_1(x)
=
\int_{a_2}^{b_2}\overline{\widetilde H_*(\xi,x)}\,e_\eta(x)\,d\sigma_2(x),
]
so that to every function (f(x)\in H_1) there corresponds a certain function (g(x)\in H_2) such that
[
\int_{a_1}^{b_1} f(x)e_\xi(x)\,d\sigma_1(x)
=
\int_{a_2}^{b_2}\overline{\widetilde H_*(\xi,x)}\,g(x)\,d\sigma_2(x),
]
and moreover
[
\int_{a_2}^{b_2} g(x)e_\xi(x)\,d\sigma_2(x)
=
\int_{a_1}^{b_1} f(x)\overline{\widetilde K_*(\xi,x)}\,d\sigma_1(x).
]
This definition is justified by the following considerations. Let (\widetilde K_n(x)) and (\widetilde K_n^*(x)) be two complete sequences of functions in (L_2(a,b)), forming a biorthogonal system.
Put
[
\widetilde K(\xi,x)=\int_0^\infty \widetilde K_u(x)e_\xi(x)\,d[u],
\qquad x\in(a,b),\quad \xi=1,2,\ldots,
]
[
\widetilde H_(\xi,x)=\int_a^b \widetilde K_x^(u)e_\xi(u)\,du,
\qquad \xi\in(a,b),\quad x=1,2,\ldots .
]
It is easy to see that if ({\widetilde K_n(x)}) is a basis, then for (\widetilde K(\xi,x)) and (\widetilde H_*(\xi,x)) all the assertions hold which were adopted in the above definition of Riesz kernels.
Theorem 10. Every (B)-kernel (\widetilde K(\xi,x)) that is a Riesz—Fischer kernel is also a Riesz kernel. At the same time the adjoint kernel (\widetilde K_*(\xi,x)) is also a Riesz kernel.
In conclusion we note that, as was established in ((^1)) (Theorem 3), if the operators (R_1) and (R_{*1}) form an isometric pair, then each of them generates a Riesz kernel in the sense of the definition given above, and these kernels are adjoint to one another.
Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR
Received
25 II 1960
References
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