Two sharp inequalities for operators in a Hilbert space

. In this paper we obtained generalisations of the L. V. Tai-kov’s and N. Ainulloev’s sharp inequalities, which estimate a norm of function’s ﬁrst-order derivative (L. V. Taikov) and a norm of function’s second-order derivative (N. Ainulloev) via the modulus of continuity or the modulus of smoothness of the function itself and the modulus of continuity or the modulus of smoothness of the function’s second-order derivative. The generalisations are obtained on the power of unbounded self-adjoint operators which act in a Hilbert space. The moduli of continuity or smoothness are deﬁned by a strongly continuous group of unitary operators.


Introduction
In 1976 L. V. Taikov obtained (see [7]) sharp inequalities, which estimate a norm of function's derivative in terms of the modulus of continuity of function itself and the modulus of continuity of its second derivative for the cases of 2π-periodic functions and functions, which are defined on real line, in space L p , 1 p < ∞. Later, in 1991, N. Ainulloev under the same conditions obtained (see [1]) sharp inequalities for a norm of function's second derivative in terms of the second-order modulus of smoothness of the function itself and the secondorder modulus of smoothness of the function's second derivative. In this paper we obtain generalisations of mentioned results in the case when operators of (multiple) differentiation are substituted by powers of unbounded self-adjoint operators acting in a Hilbert space. The modulus of continuity and the modulus of smoothness of the elements of a Hilbert space are defined in terms of a strongly continuous group of unitary operators generated by this self-adjoint operator.
The paper has the following structure. In Section 2 we present necessary notations, some facts from spectral theory, definitions of the modulus of continuity and modulus of smoothness for elements of a Hilbert space, some properties of the Riemann integral for abstract functions. The main results of the paper are contained in Section 3.

Some necessary facts
Let H be a Hilbert space over C with the inner product (f, g) H = (f, g), f, g ∈ H, and norm f H = f := (f, f ) 1 2 . Let also L(H) be a space of continuous linear operators G : H → H. By a partition of unity (for details see, for example, Chapter VI §67 in [2], Chapter XIII §1 in [5]) we mean a one-parameter family of projecting operators E s ∈ L(H), s ∈ R, satisfying the following properties: Further, let A be a self-adjoint operator with domain D A dense in H. It is well known (Chapter VI §75 in [2], Chapter XIII §6 in [5]), that each self-adjoint operator A has a uniquely defined partition of unity E s , s ∈ R, and for all f ∈ D A there exists a spectral decomposition with domain D A = f ∈ H : Af 2 < ∞ and the square of the norm Also (see [2], p. 252), in order for n-fold application of the self-adjoint operator A to be admitted to some f ∈ H, it is necessary and sufficient that the inequality holds, and if inequality (3) holds, then next equalities are true It is also known (see Chapter VI §73 in [2], Chapter XIII §7 in [5]), that the selfadjoint operator A corresponding to the partition of unity E s generates a oneparameter unitary group U t ∈ L(H), t ∈ R, i.e., an operator-valued function defined on R which values are unitary operators in H with the representation and the following properties hold: The considered strongly continuous unitary group, being one of the generalisations of the shift operator, allows one to construct for the elements of H such characteristics as the modulus of continuity and the modulus of smoothness. An overview of the results obtained in this direction and further references can be found in [6]. Let us define in H the operators of the k-th symmetric difference with step t, t ∈ R, k ∈ N: In particular and for each f ∈ H By the modulus of smoothness of order k, k ∈ N, of an element f ∈ H we mean the quantity for k = 1 we have in (6) the modulus of continuity of an element f ∈ H.
In conclusion of this section, we note some properties of the Riemann integral of an abstract function x(t), continuous on the interval [a, b] with values in H (for more details see, for example, Chapter VI §25 in [8]), which we will need along with such natural properties as the fairness of Newton-Leibniz formula and integration by parts:

Two sharp inequalities
In this section we prove two following theorems, which generalise results of [7] and [1]. Some methods, which are using next, are based on spectral decomposition and were considered, for instance, in [3], [4]. Theorem 1. For each f ∈ D A 2 and real λ > 0 the inequality holds and, hence, the second one also holds. If operator A is such, that for any real u, v, 0 < u < v λ the condition is satisfied (see, for example, [3], [4]), then for any λ > 0 both inequalities are sharp in the sense, that the inequalities stop to be true for a certain f ∈ D A 2 , if the right-hand side of (7), (8) is multiplied by a factor (1 − τ ) for arbitrary small τ > 0.
To prove sharpness of inequality (8) it is sufficiently to notice that follow equalities are true: (we used monotony of functions ∆ 1 2t g λ, and ∆ 1 2t A 2 g λ, by argument t (see corresponding representation in (18) and (16))).
Theorem 2. For each f ∈ D A 2 and real λ > 0 the inequality holds and, hence, the second one Inequalities (20) and (21) become sharp under the same condition (9) as in Theorem 1.
Proof. The ideas of the previous proof are applied here. Let us consider the operator λ > 0, and estimate each component of the right-hand side of the inequality We have To estimate the second component we applied integration by parts twice: Summing up the obtained results, we have inequality (20), and, hence, taking into account (6), estimate (21) follows. Sharpness of inequality (20) is proved in the same way as in the previous case. Let condition (9) be satisfied. Assume, that there exists τ ∈ (0, 1) such that for any f ∈ D A 2 , λ > 0, 0 ∆ 2 t f sin λtdt .