Let H be a Hilbert space in which a symmetric operator S with a dense domain Ds is given and let S have a finite deficiency index (r, s). This paper contains a necessary and sufficient condition for validity of the following inequalities of Kolmogorov type
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and a method for calculating the best possible constants Cn,m(S).
Moreover, let φ be a symmetric bilinear functional with a dense domain Dφ such that Ds ⊂ Dφ and φ(f, g) = (Sf, g) for all f ∈ Ds, g ∈ Dφ. A necessary and sufficient condition for validity of the inequality
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as well as a method for calculating the best possible constant K are obtained. Then an analogous approach is worked out in order to obtain the best possible additive inequalities of the form
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The paper is concluded by establishing the best possible constants in the inequalities
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where T is an arbitrary dissipative operator. The theorems are extensions of the results of Ju. I. Ljubič, W. N. Everitt, and T. Kato.