  In this paper we extensively investigate the class of conditionally positive
definite operators, namely operators generating conditionally positive definite
sequences. This class itself contains subnormal operators, $2$- and
$3$-isometries and much more beyond them. Quite a large part of the paper is
devoted to the study of conditionally positive definite sequences of
exponential growth with emphasis put on finding criteria for their positive
definiteness, where both notions are understood in the semigroup sense. As a
consequence, we obtain semispectral and dilation type representations for
conditionally positive definite operators. We also show that the class of
conditionally positive definite operators is closed under the operation of
taking powers. On the basis of Agler's hereditary functional calculus, we build
an $L^{\infty}(M)$-functional calculus for operators of this class, where $M$
is an associated semispectral measure. We provide a variety of applications of
this calculus to inequalities involving polynomials and analytic functions. In
addition, we derive new necessary and sufficient conditions for a conditionally
positive definite operator to be a subnormal contraction (including a
telescopic one).
