Integral Equations and Operator Theory

Abstract A one to one correspondence is established between a class of positive real rational functions with finite dimensional Naimark realizations and the set of sequences of pairs consisting of a strictly positive operator and a skew symmetric operator on a non-increasing sequence of subspaces.

Abstract In this paper we continue our investigation of unbounded Toeplitz operators whose symbols are rational matrix valued functions with poles on the unit circle, which was initiated in [16]. We further develop state space realization techniques for the symbols of these unbounded Toeplitz operators, and use the techniques to get more concrete results on the action, kernel and range, as well a…

Abstract Let $$\mathcal {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> be a $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebra. We say that $$\mathcal {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> satisfie…

Abstract We establish operator-valued versions of the earlier foundational factorization results for noncommutative polynomials due to Helton (Ann. Math., 2002) and one of the authors (Linear Alg. Appl., 2001). Specifically, we show that every positive operator-valued noncommutative polynomial p admits a single-square factorization $$p=r^{*}r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/Mat…

Abstract Using a generalized Birman–Schwinger principle developed in [31] for operators formally given by $$H_0 + V$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> and the theory of Sobolev multipliers, we develop Birman–Schwinger principles for the fol…

Abstract We develop the theory of minimal realizations and factorizations of rational functions where the coefficient space is a ring of the type introduced in our previous work, the scaled quaternions, which includes as special cases the quaternions and the split quaternions. The methods involved are not a direct generalization of the complex or quaternionic settings, and in particular, the adjo…

Abstract Let $$0&lt;p&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> and $$\Psi : [0,1) \rightarrow (0,\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>:</mml:mo> <mml:mo>[</mml:mo>…

Abstract Let $$\mathscr {H}^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> be the set of all Dirichlet series $$\textstyle f\!=\!{{\sum \limits _{n=1}^\infty }} a_nn^{-s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:mrow> <mml:mi>f</mml:mi> <mml:mspace/> <mml:mo>=</mml:mo…

Abstract We establish left and right canonical factorizations of Hilbert-space operator-valued functions G ( z ) that are analytic on neighborhoods of the complex unit circle $${\mathbb {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> and the origin 0 and that have the form $$G(z)=I+F(z)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mm…

Abstract We study quantum harmonic analysis (QHA) on the Bergman space $$\mathcal {A}^2(\mathbb {B}^n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mr…