Application of spectral decomposition to establish inequalities for operators

. We give speci(cid:28)c examples of the spectral decomposition of self-adjoint operators in application to establish sharp inequalities for their powers

In 2010, V. Babenko and R. Bilichenko [2] generalized above results for arbitrary powers of self-adjoint operator acting in Hilbert space: Let H be a Hilbert space, A be a linear, unbounded, self-adjoint operator in H, and let D(A) be its domain of denition, f be an arbitrary linear functional, r, k ∈ N, k < r. For all x ∈ D (A r ) and any τ > 0 sharp inequality holds E t is the resolution of the identity corresponding to the operator A. This will be written later.
The multiplicative inequality is proposed in the same paper. If for α, z > 0 then the following sharp inequality in multiplicative form holds

Information on the spectral theory of operators
The following supporting information can be found in more detail, for A resolution of the identity is a one-parameter family of projection operators E t dened on a nite or an innite segment [α, β] and satisfying the following conditions: 2. in the sence of stronge convergence, one has E t−0 = E t ; 3. E α = 0, E β = I.

IV
According to the spectral theorem, every self-adjoint operator A is associated with resolution of the identity E t , t ∈ R. The equality holds for all where the integral is an operator Stieltjes integral. And Using the spectral decomposition (2.1) for x ∈ D (A r ) and we will have the functional f The purpose of our publication is to nd out the resolution of the identity for specic examples of self-adjoint operators. In order to obtain new inequalities for operators based on the above results.

Spectral decomposition and inequalities for degrees of specic self-adjoint operators
Dierentiation operator in L r 2,2 (R). As the operator present in inequalities (1.3) and (1.4), we consider the operator Ax(t) = ix (t). For s < t the resolution of the identity corresponding to the operator A can be represented in the form (see [1]): 2ε as the functional in inequality (1.4). The following estimate is true: Dierentiation operator in L r 2,2 (T). Now consider the dierentiation operator Ax(t) = ix (t) acting in L 2 (T). The corresponding resolution of the identity is determined by equality: x(n) is the n-th Fourier coecient for x: We choose for m ∈ N f m (t) = m n=−m e int , that is, in fact, the Dirichlet kernel D m (t). Equality holds for such a functional: 1 2π Laplace operator in L 2 (R ). Let us also consider the Laplace operator: It is known (see,e.g., [7]) the resolution of the identity corresponding to this operator is characterized by the relation: (the series on the right side is absolutely convergent).
For ε > 0 dene a functional f ε (t) = 1 (2πε) m/2 e − |z| 2 2ε , z ∈ R m . Given the (3.4), equality estimates for the Bessel function, one can show that for all z > 0 By substituting this estimates into inequality (1.4) written for the functional f ε , we obtain for x ∈ L r 2,2 (R m ) and ε > 0 .
as ε → 0, from (3.6) we get inequality for estimating the norm of the Laplace .

Conclusion
In