Isometries on Orthogonal Sums of Kreĭn Spaces

Abstract This paper investigates isometries on orthogonal direct sums of Kreĭn spaces and subspaces. Internal sums (i-sums) of pairwise orthogonal regular subspaces $$\mathfrak {K}_n$$ K n , $$n\ge 0$$ n ≥ 0 of a Kreĭn space $$\mathfrak {K}$$ K associated with given fundamental symmetries $$J_n$$ J n , $$n\ge 0$$ n ≥ 0 are introduced as subspaces isomorphic with external sums. A characterization of multiplication operators between external and internal sums is provided in terms of the uniform norm boundedness of the restrictions to the components. We introduce internal shifts (i-shifts) as operators unitarily equivalent to standard shifts on Hardy-type Kreĭn spaces and establish several equivalent characterizations. A Wold-type decomposition theorem for isometries on Kreĭn spaces is proved, decomposing an isometry into a unitary part and an i-shift under suitable regularity conditions. We study (minimal) unitary extensions U on a Kreĭn space $$\mathfrak {K}$$ K of bounded isometries acting on a Kreĭn subspace $$\mathfrak {H}$$ H . The regularity of the minimal subspace $$\bigvee _{n\le 0}U^n\mathfrak {H}$$ ⋁ n ≤ 0 U n H is characterized in terms of the convergence of $$(U^{*n}PU^n)_{n\ge 0}$$ ( U ∗ n P U n ) n ≥ 0 , where P is the orthogonal projection onto $$\mathfrak {H}$$ H . Conditions for $$U^*|_{\mathfrak {K}\ominus \mathfrak {H}}$$ U ∗ | K ⊖ H to be a shift or an i-shift are established. We study the possibility that shifts on Kreĭn spaces have minimal unitary extensions of bilateral shift type. We prove, for example, that V and $$U^*|_{\mathfrak {K}\ominus \mathfrak {H}}$$ U ∗ | K ⊖ H