where S is a unilateral forward shift of infinite multiplicity, S∗ is its adjoint, and Yα is a matrix (αi+jCi+j ), with {αk}k∈N a sequence of complex numbers, and {Ck}k∈N a family of operators satisfying the “canonical anticommutation relations”. We exhibit a sequence {αk} such that T is polynomially bounded but T⊗T is not. This shows that the product of two commuting polynomially bounded opera...

We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general model-theoretical investigations on fragments of bounded arithmetic.

In this article we formalize negligible functions that play an essential role in cryptology [10], [2]. Generally, a cryptosystem is secure if the probability of succeeding any attacks against the cryptosystem is negligible. First, we formalize the algebra of polynomially bounded sequences [20]. Next, we formalize negligible functions and prove the set of negligible functions is a subset of the ...

(1) ‖p(T )‖ ≤M sup{|p(ζ)| : |ζ| = 1} ∀ polynomial p, and to be power bounded (notation T ∈ (PW)) if (1) holds for every polynomial of the special form p(ζ) = ζ where n is a positive integer. If T ∈ (PB) [resp., T ∈ (PW)], then there is a smallest number M which satisfies (1) [resp., (1) restricted]. This number will be called the polynomial bound of T [resp., the power bound of T ] and denoted ...