Power Series Spaces Research Articles

Isomorphic classification of Cartesian products of power series spaces.

Isomorphic classification of Cartesian products of power series spaces.

A generalization of the method of least squares for operator equations in some Frechet spaces

The classical method of least squares is extended to equations with an operator between Frechet spaces. Approximate solutions are obtained by minimizing the discrepancy relative to a metric, which in the Hilbert space case coincides with the metric induced by the scalar product. The convergence of a sequence of approximate solutions to the exact solution is proved. A concrete realization of the results obtained is given in the case of continuously invertible and so-called tamely invertible operators that map Frechet spaces of power series of finite and infinite type, Frechet spaces of rapidly decreasing sequences and the Frechet spaces of analytic functions given in Stein's monograph to themselves.

The de Branges-Rovnyak model with finite-dimensional coefficients

A characterization in terms of the canonical model spaces of L. de Branges and J. Rovnyak is obtained for Hilbert spaces of formal power series with vector coefficients which satisfy a difference-quotient inequality, thereby extending the closed ideal theorems of A. Beurling and P. D. Lax.

A rank theorem for analytic maps between power series spaces

A rank theorem for analytic maps between power series spaces

A construction of unitary linear systems

A construction is made of a unitary linear system whose transfer function is a given power seriesB(z) with operator coefficients such that multiplication byB(z) is an everywhere defined transformation in the space of square summable power series with vector coefficients. A condition is also given for the existence of an observable linear system with such a transfer function. For both constructions properties of the spaces are given which imply essential uniqueness of linear systems with given transfer functions. A canonical conjugate-isometric linear system is uniquely determined by its transfer function whenever the state space is a Pontryagin space.

Q-difference intertwining operators for Uq(sl(n)): general setting and the case n=3

We construct representations $\hat\pi_{\br}$ of the quantum algebra $U_q(sl(n))$ labelled by $n-1$ complex numbers $r_i$ and acting in the space of formal power series of $n(n-1)/2$ non-commuting variables. These variables generate a flag manifold of the matrix quantum group $SL_q(n)$ which is dual to $U_q(sl(n))$. The conditions for reducibility of $\hat\pi_{\br}$ and the procedure for the construction of the $q$ - difference intertwining operators are given. The representations and $q$ - difference intertwining operators are given in the most explicit form for $n=3$. In the Note Added some general results for arbitrary $n$ are given.

Q-difference intertwining operators for a Lorentz quantum algebra

Representations π̂r,r̄ of a Lorentz quantum algebra 𝒰 are constructed. They are labeled by two complex numbers r,r̄ and act in the space of formal power series of two noncommuting variables η,η̄. These variables are built from elements of the matrix Lorentz quantum group ℒ which is dual to 𝒰. The conditions for reducibility of π̂r,r̄ are given. q-difference intertwining operators in η, η̄, which realize partial equivalences of the representations π̂r,r̄ , are constructed explicitly. The whole construction is a generalization of a known procedure for q=1. The case when q is a root of unity is also considered in detail.

Legendre polynomials and the elliptic genus

The famous Legendre polynomials are intricately connected with the theory of elliptic integrals and with the formal groups associated to Jacobi quartic curves Y2 = 1 26X2 + &X4. For instance, Honda [4] has obtained new congruences for these polynomials using his theory of formal groups. Congruences modulo p2 involving Legendre polynomials are at the root of the construction of elliptic cohomology [S]. On the other hand, Legendre polynomials may be interpreted as matrix coefficients for finite-dimensional representations of the algebraic group X(2) [3]. In this article, we use well-known facts about the reduction modulo p of such representations to find a representation-theoretic statement which implies the Schur congruences for the Legendre polynomials (see Section 1). We use similar methods to obtain a new derivation for some of Honda’s congruences. In Section 3, we prove a statement about the reduction modulo a large prime number p of a certain representation of an orthogonal group to obtain a congruence (Corollary 3.2) for Gegenbauer polynomials. We also give a congruence for Tchebycheff polynomials (Proposition 3.4). The expression of the Legendre polynomials as matrix coefficients allows us to write down an intriguing formula (Proposition 2.1) for the logarithm of the formal group associated to a Jacobi quartic. This formula shows that this formal power series is an “expectation value” (in the sense of quantum theory) for the representation of GL(2) in the space of formal power series in two variables X and Y. The role of the vacuum vector is played by the formal power series exy. The parameter t for the formal power series appears as the entry of a scalar (2, 2)-matrix. This suggests that some

Isomorphy classes of spaces of holomorphic functions on open polydiscs in dual power series spaces

Let Λ_R(α) be a nuclear power series space of finite or infinite type with lim_{j→∞} (1/j) log α_j = 0. We consider open polydiscs D_a in Λ_R(α)'_b with finite radii and the spaces H(D_a) of all holomorphic functions on D_a under the compact-open topology

The de Branges-Rovnyak model

A characterization, extending results of A. Beurling, L. de Branges and J. Rovnyak, of those Hilbert spaces of formal power series, which are isometrically equal to a de Branges-Rovnyak (scalar-valued) function space H ( b ) \mathcal {H}(b) , is obtained.

Imbedding of power series spaces and spaces of analytic functions

The diametral dimension of a nuclear Frechet spaceE, which satisfies (DN) and (Ω), is related to power series spaces Λ1(e) and Λ∞(e) for some exponent sequence e. It is proved thatE contains a complemented copy of Λ∞(e) provided the diametral dimensions ofE and Λ∞(e) are equal and e is stable. Assuming Λ1(e) is nuclear, any subspace of Λ1(e) which satisfies (DN), can be imbedded intoE. Applications of these results to spaces of analytic functions are given.

Power series space representations of nuclear Fréchet spaces

Let E E be a nuclear graded Fréchet space such that the norms satisfy inequalities | | | | k 2 ≤ C k | | | | k − 1 | | | | k − 1 ||||_k^2 \leq {C_k}|||{|_{k - 1}}|||{|_{k - 1}} for all k k , let F F be a graded Fréchet space such that the dual (extended real valued) norms satisfy inequalities | | | | k ∗ 2 ≤ D k | | | | k − 1 ∗ | | | | k + 1 ∗ ||||_k^{*2} \leq {D_k}||||_{k - 1}^*||||_{k + 1}^* for all k k , and let A A be a tame (resp. linearly tame) linear map from F F to E E . Then there exists a tame (resp. linearly tame) factorization of A A through a power series space Λ ∞ 2 ( α ) \Lambda _\infty ^2(\alpha ) . In the case of a tame quotient map, E E is tamely equivalent to a power series space of infinite type. This applies in particular to the range of a tame (resp. linearly tame) projection in a power series space Λ ∞ 2 ( α ) \Lambda _\infty ^2(\alpha ) . In this case one does not need nuclearity. It also applies to the tame spaces in the sense of the various implicit function theorems. If they are nuclear, they are tamely equivalent to power series spaces Λ ∞ ( α ) {\Lambda _\infty }(\alpha ) .

On Projections in Power Series Spaces and the Existence of Bases

Mityagin posed the problem, whether complemented subspaces of nuclear infinite type power series spaces have a basis. A related more general question was asked by Pełczyński. It is well known for a complemented subspace $E$ of a nuclear infinite type power series space, that its diametral dimension can be represented by $\Delta E = \Delta {\Lambda _\infty }(\alpha )$ for a suitable sequence $\alpha$ with ${\alpha _j} \geq \ln (j + 1)$. In this article we prove the existence of a basis for $E$ in case that ${\alpha _j} \geq j$ and $\sup \tfrac {{{\alpha _{2j}}}}{{{\alpha _j}}} < \infty$.

On projections in power series spaces and the existence of bases

Mityagin posed the problem, whether complemented subspaces of nuclear infinite type power series spaces have a basis. A related more general question was asked by Pełczyński. It is well known for a complemented subspace E E of a nuclear infinite type power series space, that its diametral dimension can be represented by Δ E = Δ Λ ∞ ( α ) \Delta E = \Delta {\Lambda _\infty }(\alpha ) for a suitable sequence α \alpha with α j ≥ ln ⁡ ( j + 1 ) {\alpha _j} \geq \ln (j + 1) . In this article we prove the existence of a basis for E E in case that α j ≥ j {\alpha _j} \geq j and sup α 2 j α j &gt; ∞ \sup \tfrac {{{\alpha _{2j}}}}{{{\alpha _j}}} &gt; \infty .

On the minimal space of surjectivity question for linear transformations on vector spaces with applications to surjectivity of differential operators on locally convex spaces

We use transfinite induction to show that if L is an epimorphism of a vector space V and maps a vector subspace W of V into a proper subspace of itself, then there is a smallest subspace E of V containing W such that L(E) E (or a minimal space of surjectivity or solvability) and we give examples where there are infinitely many distinct minimal spaces of solvability. We produce an example showing that if L 1 and L 2 are two epimorphisms of a vector space V which are endomorphisms of a proper subspace W of V such that LI(W) n L2(W is a proper subspace of W, then there may not exist a smallest subspace E of V containing W such that LI(E) E L2(E). While no nonconstant linear partial differential operator maps the field of meromorphic functions onto itself, we construct a locally convex topological vector space of formal power series containing the meromorphic functions such that every linear partial differential operator with constant coefficients maps this space linearly and continuously onto itself. Furthermore, we show that algebraically there is for every linear partial differential operator P(D) with constant coefficients a smallest extension E of the meromorphic functions in n complex variables, where

Complemented subspaces in tame power series spaces

Complemented subspaces in tame power series spaces

On the characterization of subspaces and quotient spaces of stable power series spaces of finite type

On the characterization of subspaces and quotient spaces of stable power series spaces of finite type

On a class of suboptimal controls for infinite-dimensional bilinear systems

The suboptimal control of a bilinear system is considered with respect to a quadratic cost criterion. The feedback control is in the space of formal power series on a Hilbert space.

Tameness of Pairs of Nuclear Power Series Spaces and Related Topics

Tameness of Pairs of Nuclear Power Series Spaces and Related Topics

Tameness of pairs of nuclear power series spaces and related topics

The equivalence of the following six assertions is proved: (i) The set of the finite limit points of the ratios α m / β n , n , m ∈ N {\alpha _m}/{\beta _n},n,m \in {\mathbf {N}} , is bounded, (ii) Every operator from Λ ∞ ( β ) {\Lambda _\infty }(\beta ) to Λ 1 ( α ) {\Lambda _1}(\alpha ) is compact, (iii) The pair ( Λ ∞ ( β ) , Λ 1 ( α ) ) ({\Lambda _\infty }(\beta ),\,{\Lambda _1}(\alpha )) is tame, i.e., for every operator T T from Λ ∞ ( β ) {\Lambda _\infty }(\beta ) to Λ 1 ( α ) {\Lambda _1}(\alpha ) there is a positive integer a a such that for every k ∈ N k \in {\mathbf {N}} there is a constant C k {C_k} such that | | T x | | k ⩽ C k | x | a k ||Tx|{|_k} \leqslant {C_k}|x{|_{ak}} for every x ∈ Λ ∞ ( β ) x \in {\Lambda _\infty }(\beta ) . (iv) Every short exact sequence 0 → Λ τ ( β ) → X → Λ 1 ( α ) → 0 0 \to {\Lambda _\tau }(\beta ) \to X \to {\Lambda _1}(\alpha ) \to 0 , where τ = 1 \tau = 1 or ∞ \infty , splits. (v) Λ 1 ( α ) × Λ ∞ ( β ) {\Lambda _1}(\alpha ) \times {\Lambda _\infty }(\beta ) has a regular basis, (vi) Λ 1 ( α ) ⊗ Λ ∞ ( β ) {\Lambda _1}(\alpha ) \otimes {\Lambda _\infty }(\beta ) has a regular basis. Also the finite type power series spaces that contain subspaces isomorphic to an infinite type power series space are characterized as well as the infinite type power series spaces that have finite type quotient spaces.