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Defining conditions for irreducible tensor operators associated with theunitary irreducible corepresentations of compact quantum group algebras arededuced within the framework of the abstract carrier space formalism. It isshown that there are {\em{two}} types of irreducible tensor operator, which maybe called `ordinary' and `twisted'. The consistency of the definitions isdemonstrated, and various consequences are deduced, including generalizationsof the Wigner-Eckart theorem for both the ordinary and twisted operators.Examples of irreducible tensor operators for the standard deformation of thefunction algebra of the compact Lie group $SU(2)$ are described to demonstratethe applicability of the new definitions.Comment: To be published in J.Math.Phys., 32 pages, RevTe

Abstract carrier space formalism for the irreducible tensor operators of compact quantum group algebras