Abstract.A closed subspace H 2 (D 2 ) is said to be invariant if it is invariant under the Toeplitz operators T z and T w . Invariant subspaces of H 2 (D 2 ) are well-known to be very complicated. So discovering some good examples of invariant subspaces will be beneficial to the general study. This paper studies a type of invariant subspace constructed through a sequence of inner functions. It will be shown that this type of invariant subspace has direct connections with the Jordan operator. Related calculations also give rise to a simple upper bound for j 1 − |λ j |, where {λ j } are zeros of a Blaschke product.

Abstract. In this paper we shall give an affirmative solution to a problem, posed by Berger, Coburn and Lebow, for C * -algebras on Hardy submodules.

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In this paper, we introduce the notion of reproducing kernel Hilbert spaces for graphs and the Gram matrices associated with them. Our aim is to investigate the Gram matrices of reproducing kernel Hilbert spaces. We provide several bounds on the entries of the Gram matrices of reproducing kernel Hilbert spaces and characterize the graphs which attain our bounds.

Abstract. Let N be an integer which is larger than one. In this paper we study invariant subspaces of L 2 (T N ) under the double commuting condition. A main result is an N-dimensional version of the theorem proved by Mandrekar and Nakazi. As an application of this result, we have an N-dimensional version of Lax's theorem.