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Linear fractional maps of the ball and their composition operators
 Acta Sci. Math. (Szeged
"... In this paper, we describe a class of maps of the unit ball in CN into itself that generalize the automorphisms and deserve to be called linear fractional maps. They are special cases or generalizations of the linear fractional maps studied by Kren and Smul’jan, Harris and others. As in the complex ..."
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In this paper, we describe a class of maps of the unit ball in CN into itself that generalize the automorphisms and deserve to be called linear fractional maps. They are special cases or generalizations of the linear fractional maps studied by Kren and Smul’jan, Harris and others. As in the complex plane, a linear fractional map on CN is represented by an (N+1)(N+1) matrix. Basic connections between the properties of the map and the properties of this matrix viewed as a linear transformation on an associated Kren space are established. These maps are shown to induce bounded composition operators on the Hardy spaces Hp(BN) and some weighted Bergman spaces and we compute the adjoints of these composition operators on these spaces. Finally, we solve Schroeder’s equation f ’ = ’0(0)f when ’ is a linear fractional selfmap of the ball xing 0.
Unbounded symmetric homogeneous domains in spaces of operators, Ann
 SINGULARITIES IN BANACH ALGEBRAS 13
"... We define the domain of a linear fractional transformation in a space of operators and show that both the affine automorphisms and the compositions of symmetries act transitively on these domains. Further, we show that Liouville’s theorem holds for domains of linear fractional transformations and, w ..."
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Cited by 2 (1 self)
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We define the domain of a linear fractional transformation in a space of operators and show that both the affine automorphisms and the compositions of symmetries act transitively on these domains. Further, we show that Liouville’s theorem holds for domains of linear fractional transformations and, with an additional trace class condition, so does the Riemann removable singularities theorem. We also show that every biholomorphic mapping of the operator domain I < Z ∗ Z is a linear isometry when the space of operators is a complex Jordan subalgebra of L(H) with the removable singularity property and that every biholomorphic mapping of the operator domain I + Z ∗ 1 Z1 < Z ∗ 2 Z2 is a linear map obtained by multiplication on the left and right by Junitary and unitary operators, respectively. 0. Introduction. This paper introduces a large class of finite and infinite dimensional symmetric affinely homogeneous domains which are not holomorphically equivalent to any bounded domain. These domains are subsets of spaces
Geometric properties of linear fractional maps, preprint
, 2004
"... Abstract. Linear fractional maps in several variables generalize classical linear fractional maps in the complex plane. In this paper, we describe some geometric properties of this class of maps, especially for those linear fractional maps that carry the open unit ball into itself. For those linear ..."
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Cited by 2 (0 self)
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Abstract. Linear fractional maps in several variables generalize classical linear fractional maps in the complex plane. In this paper, we describe some geometric properties of this class of maps, especially for those linear fractional maps that carry the open unit ball into itself. For those linear fractional maps that take the unit ball into itself, we determine the minimal set containing the open unit ball on which the map is an automorphism which provides a means of classifying these maps. Finally, when ’ is a linear fractional map, we describe the linear fractional solutions, f, of Schroeder’s functional equation f ’ = Lf. 1.
Factorizations of operator matrices
 Linear Algebra Appl
"... This note gives explicit factorizations of a 2×2 operator matrix as a product of an upper triangular operator matrix and an involutory, unitary or Junitary operator matrix. A pattern is given for construction of factorizations of this kind. Let H and K be Hilbert spaces and let L(H, K) denote the s ..."
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Cited by 1 (1 self)
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This note gives explicit factorizations of a 2×2 operator matrix as a product of an upper triangular operator matrix and an involutory, unitary or Junitary operator matrix. A pattern is given for construction of factorizations of this kind. Let H and K be Hilbert spaces and let L(H, K) denote the space of all bounded[ linear operators] from H to K. Put L(H) = L(H, H). Throughout, A B M = denotes any given operator in L(K × H). To describe
Computation of Functions of Certain Operator Matrices
"... This note gives a simple method to compute the entries of holomorphic functions of a 2 \Theta 2 block or operator matrix which can be written as a product. To illustrate this method, the entries are given for the exponential, fractional powers and inverse of such operator matrices. If H and K are Hi ..."
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This note gives a simple method to compute the entries of holomorphic functions of a 2 \Theta 2 block or operator matrix which can be written as a product. To illustrate this method, the entries are given for the exponential, fractional powers and inverse of such operator matrices. If H and K are Hilbert spaces, we denote the space of bounded linear operators from H to K by L(H;K ). Our main result is the following: Theorem 1 Let B 1 ; C 1 2 L(H;K 1 ) and B 2 ; C 2 2 L(H;K 2 ). Suppose f is a function which is holomorphic in an open set D containing the spectrum oe(C 1 B 1 + C 2 B 2 ) and 0. Then f( " B 1 C 1 B 1 C 2 B 2 C 1 B 2 C 2 # ) = " f(0)I +B 1 RC 1 B 1 RC 2 B 2 RC 1 f(0)I +B 2 RC 2 # ; where R = g(C 1 B 1 + C 2 B 2 ) and g is the holomorphic extension to D of g(z) = [f(z) \Gamma f(0)]=z. Note that when H = C the operators B 1 , B 2 , C 1 and C 2 may be identified with vectors in K 1 or K 2 . The proof of Theorem 1 follows immediat...
a function which is holomorphic in an open set D containing the spectrum σ(C ∗ 1B1 + C ∗ 2B2) and 0. Then
"... This note gives a simple method to compute the entries of holomorphic functions of a 2 × 2 block or operator matrix which can be written as a product. To illustrate this method, the entries are given for the exponential, fractional powers and inverse of such operator matrices. If H and K are Hilbert ..."
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This note gives a simple method to compute the entries of holomorphic functions of a 2 × 2 block or operator matrix which can be written as a product. To illustrate this method, the entries are given for the exponential, fractional powers and inverse of such operator matrices. If H and K are Hilbert spaces, we denote the space of bounded linear operators from H to K by L(H, K). Our main result is the following: