Quantum Calculus and Ideals of Compact Operators

One of the goals of noncommutative geometry is the translation of basic notions of analysis into the language of Banach algebras. This translation is done using the quantization procedure. The arising operator calculus is called, following Connes, the quantum calculus. In this paper we shall give several assertions from this calculus concerning the interpretation of Schatten ideals of compact operators in a Hilbert space in terms of function theory. The main attention is paid to the case of Hilbert–Schmidt operators.