In this dissertation, we solve multivariable Nevanlinna-Pick type interpolation problems. Particularly, we consider the left tangential interpolation problems on the commutative or noncommutative unit ball. For the commutative setting, we discuss left-tangential operator-argument interpolation problems for Schur-class multipliers on the Drury-Arveson space and for the noncommutative setting, we discuss interpolation problems for Schur-class multipliers on Fock space. We apply the Krein-space geometry approach (also known as the Grassmannian Approach). To implement this approach J-versions of Beurling-Lax representers for shift-invariant subspaces are required. Here we obtain these J-Beurling-Lax theorems by the state-space method for both settings. We see that the Krein-space geometry method is particularly simple in solving the interpolation problems when the Beurling-Lax representer is bounded. The Potapov approach applies equally well whether the representer is bounded or not. / Ph. D.

Let <i>φ</i> be a self-map of <b>B</b>₂, the unit ball in <b>C</b>². We investigate the equation <i>C_φf</i>=<i>λf</i> where we define <i>C_φf </i>: -<i>f◦φ</i>, with <i>f a</i> function in the Drury Arves on Space. After imposing conditions to keep <i>C_φ</i> bounded and well-behaved, we solve the equation <i>C_φf - λf </i>and determine the spectrum <i>σ</i>(<i>C_φ</i>) in the case where there is no interior fixed point and boundary fixed point without multiplicity. We then investigate the existence of one-parameter semigroups for such maps and discuss some generalizations.