Analytical Applications on Some Hilbert Spaces

In this paper, we establish an uncertainty inequality for a Hilbert space H . The minimizer function associated with a bounded linear operator from H into a Hilbert space K is provided. We come up with some results regarding Hardy and Dirichlet spaces on the unit disk  .


Introduction
Hilbert spaces are the most important tools in the theories of partial differential equations, quantum mechanics, Fourier analysis, and ergodicity. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of squareintegrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Saitoh et al. applied the theory of Hilbert spaces to the Tikhonov regularization problems [1,2]. Matsuura et al. obtained the approximate solutions for bounded linear operator equations with the viewpoint of numerical solutions by computers [3,4]. During the last years, the theory of Hilbert spaces has gained considerable interest in various fields of mathematical sciences [5][6][7][8][9]. We expect that the results of this paper will be useful when discussing (in Section 2) uncertainty inequality for Hilbert space H and minimizer function associated with a bounded linear operator T from H into a Hilbert space K. As applications, we consider Hardy and Dirichlet spaces as follows.
Let  be the complex plane and  ¼ z ∈  :jzj < 1 f gthe open unit disk. The Hardy space H  ð Þ is the set of all analytic functions f in the unit disk  with the finite integral: It is a Hilbert space when equipped with the inner product: Over the years, the applications of Hardy space H  ð Þ play an important role in various fields of mathematics [5,10] and in certain parts of quantum mechanics [11,12]. And this space is the background of some applications. For example, in Section 3, we study on H  ð Þ the following two operators: and we deduce uncertainty inequality for this space. Next, we establish the minimizer function associated with the difference operator: In Section 4, we consider the Dirichlet space D  ð Þ, which is the set of all analytic functions f in the unit disk  with the finite Dirichlet integral: It is also a Hilbert space when equipped with the inner product: This space is the objective of many applicable works [5,[13][14][15][16][17] and plays a background to our contribution. For example, we study on D  ð Þ the following two operators: and we deduce the uncertainty inequality for this space D  ð Þ. And we establish the minimizer function associated with the difference operator:

Generalized results
Let H be a Hilbert space equipped with the inner product :, : h i H . And let A and B be the two operators defined on H. We define the commutator A, B ½ by The adjoint of A denoted by A * is defined by for f ∈ Dom A ð Þ and g ∈ Dom A * ð Þ. Proof.
Then AA * f and A * Af belong to H.
The following result is proved in [18,19].

Theorem 2.2. Let A and B be the self-adjoint operators on a Hilbert space H. Then
where i is the imaginary unit.

Proof. Let us consider the following two operators on Dom
where The operator P given by (16) is self-adjoint; then for any real a, we have This shows that and the minimum is attained when a ¼ Similarly Then by (15), (21), and (22), we deduce the inequality (17). □ Let λ > 0 and let T : H ! K be a bounded linear operator from H into a Hilbert space K. Building on the ideas of Saitoh [2], we examine the minimizer function associated with the operator T. Theorem 2.5. For any k ∈ K and for any λ > 0, the problem has a unique minimizer given by Proof. The problem (23) is solved elementarily by finding the roots of the first derivative DΦ of the quadratic and strictly convex function Note that for convex functions, the equation DΦ f ð Þ ¼ 0 is a necessary and sufficient condition for the minimum at f . The calculation provides and the assertion of the theorem follows at once. □ Theorem 2.6. If T : H ! K is an isometric isomorphism; then for any k ∈ K and for any λ > 0, the problem has a unique minimizer given by Proof. We have T * ¼ T À1 and T * T ¼ I. Thus, by (24), we deduce the result. □

The Hardy space H  ð Þ
Let  be the complex plane and  ¼ z ∈  :jzj < 1 f gthe open unit disk. The Hardy space H  ð Þ is the set of all analytic functions f in the unit disk  with the finite integral: It is a Hilbert space when equipped with the inner product: If f , g ∈ H  ð Þ with f z ð Þ ¼ P ∞ n¼0 a n z n and g z ð Þ ¼ P ∞ n¼0 b n z n , then The set z n f g ∞ n¼0 forms an Hilbert's basis for the space H  ð Þ. The Szegő kernel S z given for z ∈ , by is a reproducing kernel for the Hardy space H  ð Þ, meaning that S z ∈ H  ð Þ, and For z ∈ , the function u z ð Þ ¼ S z w ð Þ is the unique analytic solution on  of the initial problem: In the next of this section, we define the operators ∇, ℜ, and L on H  ð Þ by These operators satisfy the commutation rule: where I is the identity operator. We define the Hilbert space U  ð Þ as the space of all analytic functions f in the unit disk  such that If f ∈ U  ð Þ with f z ð Þ ¼ P ∞ n¼0 a n z n , then n 2 a n j j 2 : Thus, the space U  ð Þ is a subspace of the Hardy space H  ð Þ. Theorem 3.1.
iii. For f ∈ U  ð Þ, one has Proof.
i. Let f ∈ U  ð Þ with f z ð Þ ¼ P ∞ n¼0 a n z n . Then and Consequently ∇f , ℜf , and Lf belong to H  ð Þ.
ii. For f , g ∈ U  ð Þ with f z ð Þ ¼ P ∞ n¼0 a n z n and g z ð Þ ¼ P ∞ n¼0 b n z n , one has Thus ∇ * ¼ L.
iii. Let f ∈ U  ð Þ. By (ii) and (34), we deduce that Theorem 3.3. Let T 1 be the difference operator defined on H  ð Þ by i. The operator T 1 maps continuously from H  ð Þ to H  ð Þ, and ii. For f ∈ H  ð Þ and z ∈ , we have iii. For any h ∈ H  ð Þ and for any λ > 0, the problem has a unique minimizer given by 6

Functional Calculus
Proof.
i. If f ∈ H  ð Þ with f z ð Þ ¼ P ∞ n¼0 a n z n , then T 1 f z ð Þ ¼ P ∞ n¼0 a nþ1 z n and ii. If f , g ∈ H  ð Þ with f z ð Þ ¼ P ∞ n¼0 a n z n and g z ð Þ ¼ P ∞ n¼0 b n z n , then where T * 1 g z ð Þ ¼ zg z ð Þ, for z ∈ . And therefore iii. From Theorem 2.5 we have is a reproducing kernel for the Dirichlet space D  ð Þ, meaning that K z ∈ D  ð Þ, and for all f ∈ D  ð Þ, we have f , K z h i D  ð Þ ¼ f z ð Þ. For z ∈ , the function u z ð Þ ¼ K z w ð Þ is the unique analytic solution on  of the initial problem: In the next of this section, we define the operators Λ, ℜ, and X on D  ð Þ by These operators satisfy the following commutation relation: We define the Hilbert space V  ð Þ as the space of all analytic functions f in the unit disk  such that n 3 a n j j 2 : Thus, the space V  ð Þ is a subspace of the Dirichlet space D  ð Þ. Theorem 4.1.
iii. For f ∈ V  ð Þ, one has Proof.
ii. For f , g ∈ V  ð Þ with f z ð Þ ¼ P ∞ n¼0 a n z n and g z ð Þ ¼ P ∞ n¼0 b n z n , one has