Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk \mathbbD{\mathbb{D}}. Each function f ? Co(a){f\in Co(\alpha)} maps the unit disk \mathbbD{\mathbb{D}} onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional (1-|z|²)( f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional (1-|z|²)(f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)} whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

The composition operators on weighted Bloch space

We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B _log = { f ? H(D): sup_{z ? D} (1-| z|²) ( log\frac21-| z|² )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.     

Iterated Function Systems and Łojasiewicz–Siciak Condition of Green’s Function

A compact set K ì \mathbbC^N{K \subset \mathbb{C}}^{N} satisfies (ŁS) if it is polynomially convex and there exist constants B,β > 0 such that

Dirichlet and Bergman Spaces of Holomorphic Functions on the Unit Ball of C n

Let B denote the unit ball in [(?)\tilde] \widetilde{\nabla\hskip-4pt}\hskip4pt denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces D_g{{\cal D}_{\gamma}} , $\gamma > (n-1)$\gamma > (n-1) , and weighted Bergman spaces A^p_a{A^p_{\alpha}} , 0 < p < ￥0 < p < \infty , $\alpha > n$\alpha > n , of holomorphic functions f on B for which D_g( f)D_{\gamma}(\,f) and || f||_A^pa\Vert\, f\Vert_{A^p_{\alpha}} respectively are finite, where D_g( f)=ò_B (1-|z|²)^g|[(?)\tilde]  f(z)|²dt(z),D_{\gamma}(\,f)=\int_B (1-\vert z\vert^2)^{\gamma}\vert\widetilde{\nabla\hskip-4pt}\hskip4pt f(z)\vert^2d\tau(z), and || f||^p_A^pa=ò_B(1-|z|²)^a| f(z)|^pdt(z).\Vert\, f\Vert^p_{A^p_{\alpha}}=\int_B(1-\vert z\vert^2)^{\alpha}\vert\, f(z)\vert^pd\tau(z). The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and $\alpha > n$\alpha > n .     

Density of the polynomials in Hardy and Bergman spaces of slit domains

It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbA^t(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in H^t(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.     

On Classes of Functions Related to Starlike Functions with Respect to Symmetric Conjugate Points Defined by a Fractional Differential Operator

Let A denote the class of analytic functions f, in the open unit disk E = {z : |z| < 1}, normalized by f(0) = f′(0) − 1 = 0. In this paper, we introduce and study the class ST^n,a_l,m(h){ST^{n,\alpha}_{\lambda,m}(h)} of functions f ? A{f\in A}, with \fracD^n,a_l f_m(z)z 1 0{\frac{D^{n,\alpha}_\lambda f_m(z)}{z}\neq 0}, satisfying

Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WF_FL^q(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FL^q_(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S^(w₀)_r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that

A new interpretation of the Shimura curve with discriminant 6 in terms of Picard modular forms

In this paper we give a realization of the Shimura curve for the quaternion algebra over Q with discriminant 6 as a quotient space of the complex upper half plane by the triangle group Δ(3, 6, 6). It is given by the Schwarz map for the Gauss hypergeometric differential equation E(\frac16,\frac13,\frac23){E\left(\frac{1}{6},\frac{1}{3},\frac{2}{3}\right)}. The corresponding abelian surfaces are obtained as isogenous components of the Jacobi varieties of the Picard curves C(s): w³=z(z-1) (z-\frac12 (1-s))(z-\frac12 (1+s)){C(s): w^3=z(z-1)\break \left(z-\frac{1}{2} (1-s)\right)\left(z-\frac{1}{2} (1+s)\right)}.     

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Smooth Universal Taylor Series

Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote S_N (g,z)(z) = ?^N_l=0\fracg^(l) (z)l ! (z-z)^lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

Dimension Free Estimates for Riesz Transforms Associated to the Harmonic Oscillator on \mathbb{R}^{n}

Let L=?Δ+|ξ|² be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|² be the harmonic oscillator on \mathbbRⁿ\mathbb{R}^{n} , with the associated Riesz transforms R_2j−1=(∂/∂ξ_j)L^−1/2,R_2j=ξ_jL^−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant C_p,

||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}.      12. Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions      M. Obradović  S. Ponnusamy  P. Vasundhra 《Proceedings Mathematical Sciences》2009,119(5):593-610 Let U(λ, μ) denote the class of all normalized analytic functions f in the unit disk |z| < 1 satisfying the condition $ \frac{{f(z)}} {z} \ne 0and\left| {f'(z)\left( {\frac{z} {{f(z)}}} \right)^{\mu + 1} - 1} \right| < \lambda ,\left| z \right| < 1. $ \frac{{f(z)}} {z} \ne 0and\left| {f'(z)\left( {\frac{z} {{f(z)}}} \right)^{\mu + 1} - 1} \right| < \lambda ,\left| z \right| < 1.       13. Toeplitz Operators on Generalized Bergman Spaces      Kamthorn Chailuek  Brian C. Hall 《Integral Equations and Operator Theory》2010,66(1):53-77 We consider the weighted Bergman spaces HL2(\mathbb Bd, ml){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dml(z) = cl(1-|z|2)l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.      14. Punishing Factors for Finitely Connected Domains      F. G. Avkhadiev  K.-J. Wirths 《Monatshefte für Mathematik》2006,106(1):103-115 Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities Cn(W,P) := supf ? A(W,P)supz ? W\frac|f(n)(z)| R(f(z),P)n! (R(z,W))nC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.      15. A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations      Xiaochun Chen  Sadek Gala  Zhengguang Guo 《Acta Appl Math》2011,113(2):207-213 In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L\frac21-r( 0,T;[(X)\dot]r( \mathbbR3) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.      16. Blow up of subcritical quantities at the first singular time of the mean curvature flow      Nam Q. Le 《Geometriae Dedicata》2011,151(1):361-371 Consider a family of smooth immersions F(·,t) : Mn? \mathbbRn+1{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò0tòMs\frac|A|n + 2 log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.      17. Numerical Ranges of Radial Toeplitz Operators on Bergman Space      Kuo Zhong Wang  Pei Yuan Wu 《Integral Equations and Operator Theory》2009,65(4):581-591 A Toeplitz operator TfT_\phi with symbol f\phi in L￥(\mathbbD)L^{\infty}({\mathbb{D}}) on the Bergman space A2(\mathbbD)A^{2}({\mathbb{D}}), where \mathbbD\mathbb{D} denotes the open unit disc, is radial if f(z) = f(|z|)\phi(z) = \phi(|z|) a.e. on \mathbbD\mathbb{D}. In this paper, we consider the numerical ranges of such operators. It is shown that all finite line segments, convex hulls of analytic images of \mathbbD\mathbb{D} and closed convex polygonal regions in the plane are the numerical ranges of radial Toeplitz operators. On the other hand, Toeplitz operators TfT_\phi with f\phi harmonic on \mathbbD\mathbb{D} and continuous on [`(\mathbbD)]{\overline{\mathbb{D}}} and radial Toeplitz operators are convexoid, but certain compact quasinilpotent Toeplitz operators are not.      18. Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems      Jaume Llibre  Clàudia Valls 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(5):657-679 In this paper we classify the centers localized at the origin of coordinates, the cyclicity of their Hopf bifurcation and their isochronicity for the polynomial differential systems in \mathbbR2{\mathbb{R}^2} of degree d that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-52 (A z4+j[`(z)]1-j + B z3[`(z)]2 + C z2-j[`(z)]3+j+D[`(z)]5), \dot z = (\lambda+i) z + (z \overline{z})^{\frac{d-5}{2}} \left(A z^{4+j} \overline{z}^{1-j} + B z^3 \overline{z}^2 + C z^{2-j} \overline{z}^{3+j}+D \overline{z}^5\right),      19. On the Convexity of the Heinz Means      Fuad Kittaneh 《Integral Equations and Operator Theory》2010,68(4):519-527 Let A, B, and X be operators on a complex separable Hilbert space such that A and B are positive, and let 0 ≤ v ≤ 1. The Heinz inequalities assert that for every unitarily invariant norm | | | ·| | | ,{\left\vert \left\vert \left\vert \cdot \right\vert \right\vert \right\vert ,} 2| | | A1/2XB1/2| | | ￡ | | | AvXB1-v+A1-vXBv| | | ￡ | | | AX+XB| | |.2\left\vert \left\vert \left\vert A^{1/2}XB^{1/2}\right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert A^{v}XB^{1-v}+A^{1-v}XB^{v}\right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert AX+XB\right\vert \right\vert \right\vert.      20. Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I      S. S. Boiko  V. K. Dubovoy 《Complex Analysis and Operator Theory》2011,5(1):157-196 Let θ(ζ) be a Schur operator function, i.e., it is defined and holomorphic on the unit disk := C : 1 {\mathbb {D} := \{\zeta \in \mathbb {C} : \vert\zeta\vert < 1 \}} and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and *-outer Schur operator functions j(z){\varphi(\zeta)} and ψ(ζ) which describe respectively the deviations of the function θ(ζ) from inner and *-inner operator functions are studied. If j(z) 1 0{\varphi(\zeta)\neq 0} , then it means that in the scattering system for which θ(ζ) is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system (Sect. 6). The function ψ(ζ) has the analogous property for the dual system. For this reason these functions are called the defect functions of the function θ(ζ). The explicit form of the defect functions j(z){\varphi(\zeta)} and ψ(ζ) is obtained and the analytic connection of these functions with the function θ(ζ) is described (Sects. 3, 5). The operator functions (l j(z)q(z)){\left(\begin{array}{l} \varphi(\zeta)\\ \theta(\zeta)\end{array}\right)} and (ψ(ζ), θ(ζ)) are Schur functions as well (Sect. 3). It is important that there exists the unique contractive measurable operator function χ(t), t ? ?\mathbb D{t\in\partial\mathbb {D}} , such that the operator function (l c(t)    j(t)y(t)    q(t) ){\left(\begin{array}{l} \chi(t)\quad \varphi(t)\\ \psi(t)\quad \theta(t) \end{array}\right)} , t ? ?\mathbb D,{t\in\partial\mathbb {D},} is also contractive (Part II, Sect. 12). The second part of the paper is devoted to studying the properties of the function χ(t). Specifically, it is shown that the function χ(t) is the scattering suboperator through the internal channels of the scattering system for which θ(ζ) is the transfer function (Part II, Sect. 12).      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions

Let U(λ, μ) denote the class of all normalized analytic functions f in the unit disk |z| < 1 satisfying the condition

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces HL²(\mathbb B^d, m_l){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dm_l(z) = c_l(1-|z|²)^l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

Blow up of subcritical quantities at the first singular time of the mean curvature flow

Consider a family of smooth immersions F(·,t) : Mⁿ? \mathbbRⁿ⁺¹{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò₀^tò_Ms\frac|A|^{n + 2} log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.     

Numerical Ranges of Radial Toeplitz Operators on Bergman Space

A Toeplitz operator T_fT_\phi with symbol f\phi in L^￥(\mathbbD)L^{\infty}({\mathbb{D}}) on the Bergman space A²(\mathbbD)A^{2}({\mathbb{D}}), where \mathbbD\mathbb{D} denotes the open unit disc, is radial if f(z) = f(|z|)\phi(z) = \phi(|z|) a.e. on \mathbbD\mathbb{D}. In this paper, we consider the numerical ranges of such operators. It is shown that all finite line segments, convex hulls of analytic images of \mathbbD\mathbb{D} and closed convex polygonal regions in the plane are the numerical ranges of radial Toeplitz operators. On the other hand, Toeplitz operators T_fT_\phi with f\phi harmonic on \mathbbD\mathbb{D} and continuous on [`(\mathbbD)]{\overline{\mathbb{D}}} and radial Toeplitz operators are convexoid, but certain compact quasinilpotent Toeplitz operators are not.     

Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems

In this paper we classify the centers localized at the origin of coordinates, the cyclicity of their Hopf bifurcation and their isochronicity for the polynomial differential systems in \mathbbR²{\mathbb{R}^2} of degree d that in complex notation z = x + i y can be written as

On the Convexity of the Heinz Means

Let A, B, and X be operators on a complex separable Hilbert space such that A and B are positive, and let 0 ≤ v ≤ 1. The Heinz inequalities assert that for every unitarily invariant norm | | | ·| | | ,{\left\vert \left\vert \left\vert \cdot \right\vert \right\vert \right\vert ,}

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

Let θ(ζ) be a Schur operator function, i.e., it is defined and holomorphic on the unit disk := C : 1 {\mathbb {D} := \{\zeta \in \mathbb {C} : \vert\zeta\vert < 1 \}} and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and *-outer Schur operator functions j(z){\varphi(\zeta)} and ψ(ζ) which describe respectively the deviations of the function θ(ζ) from inner and *-inner operator functions are studied. If j(z) 1 0{\varphi(\zeta)\neq 0} , then it means that in the scattering system for which θ(ζ) is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system (Sect. 6). The function ψ(ζ) has the analogous property for the dual system. For this reason these functions are called the defect functions of the function θ(ζ). The explicit form of the defect functions j(z){\varphi(\zeta)} and ψ(ζ) is obtained and the analytic connection of these functions with the function θ(ζ) is described (Sects. 3, 5). The operator functions (l j(z)q(z)){\left(\begin{array}{l} \varphi(\zeta)\\ \theta(\zeta)\end{array}\right)} and (ψ(ζ), θ(ζ)) are Schur functions as well (Sect. 3). It is important that there exists the unique contractive measurable operator function χ(t), t ? ?\mathbb D{t\in\partial\mathbb {D}} , such that the operator function (l c(t)    j(t)y(t)    q(t) ){\left(\begin{array}{l} \chi(t)\quad \varphi(t)\\ \psi(t)\quad \theta(t) \end{array}\right)} , t ? ?\mathbb D,{t\in\partial\mathbb {D},} is also contractive (Part II, Sect. 12). The second part of the paper is devoted to studying the properties of the function χ(t). Specifically, it is shown that the function χ(t) is the scattering suboperator through the internal channels of the scattering system for which θ(ζ) is the transfer function (Part II, Sect. 12).