Spaces Of Vector-valued Functions Research Articles

The spectral form of the functional model for maximally dissipative operators: A Lagrange identity approach

This paper is a contribution to the theory of functional models. In particular, it develops the so-called spectral form of the functional model where the selfadjoint dilation of the operator is represented as the operator of multiplication by an independent variable in some auxiliary vector-valued function space. With the help of a Lagrange identity, in the present version the relationship between this auxiliary space and the original Hilbert space will be explicit. A simple example is provided.

A simplified approach to the holomorphic discrete series

Expository article on semisimple Lie groups of Hermitian type and their unitary representations known as the holomorphic discrete series. The realization of the symmetric spaces associated to the groups as bounded symmetric domains is described. The representations in question are defined by holomorphic induction and realized on spaces of vector-valued holomorphic functions on the domain. A key question is whether the induction process yields a non-zero space. It is answered by Harish-Chandra’s condition, for which a complete proof is given.

A numerical range approach to Birkhoff–James orthogonality with applications

The main aim of this paper is to provide characterizations of Birkhoff–James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia–Šemrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of Z. As a consequence, if X is a Banach space containing strongly exposed points and Y is a smooth Banach space with dimension at least two, then there are no spear operators from X to Y. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

On the enriched mixed Galerkin method with θ scheme for the elastic wave equation

In this paper, we propose the application of an enriched Galerkin (EG) method to solve the elastic wave equation in the “displacement pseudo-pressure” mixed form, which is known for being locking-free. Unlike the continuous Galerkin (CG) method and the discontinuous Galerkin (DG) method, the EG method is based on the weak form of the DG method, and the enriched space combines a continuous piecewise linear vector-valued function space with some discontinuous piecewise linear functions. Specifically, we employ the enriched space to discretize the displacement, while the pseudo-pressure is discretized using the P0 element space. As a result, compared to the DG method, the EG method significantly reduces the number of degrees of freedom. Furthermore, we adopt a more general θ-scheme for time discretization, which behaves as an implicit scheme for 14≤θ≤1 and as an explicit scheme for 0≤θ<14. We establish error estimates for both the semi-discrete and fully-discrete schemes. The accuracy of the proposed method is validated through numerical examples, which not only confirm the theoretical analysis but also provide simulations of elastic wave propagation.

Periodic solutions of four-order degenerate differential equations with finite delay in vector-valued function spaces

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ( $P_4$ ): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$ , where $A$ , $B$ , $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$ , $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$ ) into $X$ , $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$ . We completely characterize the well-posedness of ( $P_4$ ) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

Euclidean Structures and Operator Theory in Banach Spaces

We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators Γ \Gamma on a Hilbert space. Our assumption on Γ \Gamma is expressed in terms of α \alpha -boundedness for a Euclidean structure α \alpha on the underlying Banach space X X . This notion is originally motivated by R \mathcal {R} - or γ \gamma -boundedness of sets of operators, but for example any operator ideal from the Euclidean space ℓ n 2 \ell ^2_n to X X defines such a structure. Therefore, our method is quite flexible. Conversely we show that Γ \Gamma has to be α \alpha -bounded for some Euclidean structure α \alpha to be representable on a Hilbert space. By choosing the Euclidean structure α \alpha accordingly, we get a unified and more general approach to the Kwapień–Maurey factorization theorem and the factorization theory of Maurey, Nikišin and Rubio de Francia. This leads to an improved version of the Banach function space-valued extension theorem of Rubio de Francia and a quantitative proof of the boundedness of the lattice Hardy–Littlewood maximal operator. Furthermore, we use these Euclidean structures to build vector-valued function spaces. These enjoy the nice property that any bounded operator on L 2 L^2 extends to a bounded operator on these vector-valued function spaces, which is in stark contrast to the extension problem for Bochner spaces. With these spaces we define an interpolation method, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem, we prove a transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We for example extend and refine the known theory based on R \mathcal {R} -boundedness for the joint and operator-valued H ∞ H^\infty -calculus. Moreover, we extend the classical characterization of the boundedness of the H ∞ H^\infty -calculus on Hilbert spaces in terms of B I P BIP , square functions and dilations to the Banach space setting. Furthermore we establish, via the H ∞ H^\infty -calculus, a version of Littlewood–Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our abstract setup allows us to reduce assumptions on the geometry of X X , such as (co)type and UMD. We conclude with some sophisticated counterexamples for sectorial operators, with as a highlight the construction of a sectorial operator of angle 0 0 on a closed subspace of L p L^p for 1 &gt; p &gt; ∞ 1&gt;p&gt;\infty with a bounded H ∞ H^\infty -calculus with optimal angle ω H ∞ ( A ) &gt; 0 \omega _{H^\infty }(A) &gt;0 .

Representing certain vector-valued function spaces as tensor products

Representing certain vector-valued function spaces as tensor products

Regular measures of noncompactness and Ascoli–Arzelà type compactness criteria in spaces of vector-valued functions

In this paper we estimate the Kuratowski and the Hausdorff measures of noncompactness of bounded subsets of spaces of vector-valued bounded functions and of vector-valued bounded differentiable functions. To this end, we use a quantitative characteristic modeled on a new equicontinuity-type concept and classical quantitative characteristics related to pointwise relative compactness. We obtain new regular measures of noncompactness in the spaces taken into consideration. The established inequalities reduce to precise formulas in some classes of subsets. We derive Ascoli–Arzelà type compactness criteria.

The maximal regularity property of abstract integrodifferential equations

We provide a convenient framework for the study of the well-posedness of a variety of abstract (integro)differential equations in general Banach function spaces. It allows us to extend and complement the known theory on the maximal regularity of such equations. More precisely, by methods of harmonic analysis, we identify large classes of Banach spaces which are invariant with respect to distributional Fourier multipliers. Such classes include general vector-valued Banach function spaces varPhi and/or the scales of Besov and Triebel–Lizorkin spaces defined by varPhi . We apply this result to the study of the well-posedness and maximal regularity property of abstract second-order integrodifferential equations, which model various types of elliptic and parabolic problems arising in different areas of applied mathematics.

Linear topological invariants for kernels of convolution and differential operators

We establish the condition (Ω) for smooth kernels of various types of convolution and differential operators. By the (DN)-(Ω) splitting theorem of Vogt and Wagner, this implies that these operators are surjective on the corresponding spaces of vector-valued smooth functions with values in a product of Montel (DF)-spaces whose strong duals satisfy the condition (DN), e.g., the space D′(Y) of distributions over an open set Y⊆Rn or the space S′(Rn) of tempered distributions. Most notably, we show that:(i)EP(X)={f∈E(X)|P(D)f=0} satisfies (Ω) for any differential operator P(D) and any open convex set X⊆Rd.(ii)Let P∈C[ξ1,ξ2] and X⊆R2 open be such that P(D):E(X)→E(X) is surjective. Then, EP(X) satisfies (Ω).(iii)Let μ∈E′(Rd) be such that E(Rd)→E(Rd),f↦μ⁎f is surjective. Then, {f∈E(Rd)|μ⁎f=0} satisfies (Ω). The central result in this paper is that the space of smooth zero solutions of a general convolution equation satisfies the condition (Ω) if and only if the space of distributional zero solutions of the equation satisfies the condition (PΩ). The above and related statements then follow from known results concerning (PΩ) for distributional kernels of convolution and differential operators [3,15,16].

Multivariate Gaussian processes: definitions, examples and applications

Gaussian processes occupy one of the leading places in modern statistics and probability theory due to their importance and a wealth of strong results. The common use of Gaussian processes is in connection with problems related to estimation, detection, and many statistical or machine learning models. In this paper, we propose a precise definition of multivariate Gaussian processes based on Gaussian measures on vector-valued function spaces, and provide an existence proof. In addition, several fundamental properties of multivariate Gaussian processes, such as stationarity and independence, are introduced. We further derive two special cases of multivariate Gaussian processes, including multivariate Gaussian white noise and multivariate Brownian motion, and present a brief introduction to multivariate Gaussian process regression as a useful statistical learning method for multi-output prediction problems.

Isometries between spaces of vector-valued differentiable functions II

Isometries between spaces of vector-valued differentiable functions II

Denseness of norm attaining compact operators to some vector-valued function spaces

Denseness of norm attaining compact operators to some vector-valued function spaces

Distinguished vector-valued continuous function spaces and injective tensor products

The paper continues the study on the distinguished property of the space $C_{p}\left( X\right) $ (of real-valued continuous functions over a Tychonoff space $X$ in the pointwise topology) under the formation of tensor products towards the following research directions: $\left( i\right) $ the injective tensor product $C_{p}\left( X\right) \otimes _{\varepsilon }E$ of $C_{p}\left( X\right) $ and a real locally convex space $E$ (and its completion $C_{p}\left( X\right) \,\widehat{\otimes }_{\varepsilon }\,E$), and $\left( ii\right) $ the space $C_{p}\left( X,E\right) $ of all $E$-valued continuous functions (with a normed space $E$) endowed with the pointwise topology. This work leads also to a new characterization of distinguished Fréchet locally convex spaces $E$. We show, e.g., that if $C_{p}(X)$ is metrizable, then $E$ is distinguished if and only if metrizable $C_{p}\left( X\right) \otimes _{\varepsilon }E$ is distinguished.

Stević-Sharma Operator on Spaces of Vector-Valued Holomorphic Functions

Stević-Sharma Operator on Spaces of Vector-Valued Holomorphic Functions

Conditions for compactness of integral operator on Musielak--Orlicz space of vector-valued functions

Conditions for compactness of integral operator on Musielak--Orlicz space of vector-valued functions

Banach-Stone Theorem for isometries on spaces of vector-valued differentiable functions

Let Q⊆Rm, K⊆Rn be open sets, p,q∈N, 1≤r<∞ and let E,F be Banach spaces. Denote by C⁎p(Q,E)r the space of all f∈Cp(Q,E) with bounded derivatives of order ≤p, endowed with the norm‖f‖=sups∈Q⁡‖[(‖∂λf(s)‖E)λ∈Λ]‖r, where ‖⋅‖r denotes the ℓr norm on RΛ, Λ={λ:|λ|≤p}. Let T:C⁎p(Q,E)r→C⁎q(K,F)r be a linear surjective isometry. Then m=n and p=q and there are a Cp-diffeomorphism τ:K→Q and Banach space isomorphisms V(t):E→F so thatTf(t)=V(t)f(τ(t)) if f∈C⁎p(Q,E),t∈K. The result holds in a more general setting. The proof establishes a direct link between isometries and biseparating maps.

Characterizing Sobolev spaces of vector-valued functions

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset Ω⊂RN and a Banach space V, we characterize the functions in the Sobolev-Reshetnyak space R1,p(Ω,V), where 1≤p≤∞, in terms of the existence of partial metric derivatives or partial w⁎-derivatives with suitable integrability properties. In the case p=∞ the Sobolev-Reshetnyak space R1,∞(Ω,V) is characterized in terms of a uniform local Lipschitz property. We also consider the special case of the space V=ℓ∞.

Ball covering property from commutative function spaces to non-commutative spaces of operators

A Banach space is said to have the ball-covering property (abbreviated BCP) if its unit sphere can be covered by countably many closed, or equivalently, open balls off the origin. Let K be a locally compact Hausdorff space and X be a Banach space. In this paper, we give a topological characterization of BCP, that is, the continuous function space C0(K) has the (uniform) BCP if and only if K has a countable π-basis. Moreover, we give the stability theorem: the vector-valued continuous function space C0(K,X) has the (strong or uniform) BCP if and only if K has a countable π-basis and X has the (strong or uniform) BCP. We also explore more examples for BCP on non-commutative spaces of operators B(X,Y). In particular, these results imply that B(c0), B(ℓ1) and every subspace containing finite rank operators in B(ℓp) for 1<p<∞ all have the BCP, and B(L1[0,1]) fails the BCP. Using those characterizations and results, we show that BCP is not hereditary for 1-complemented subspaces (even for completely 1-complemented subspace in operator space sense) by constructing two different counterexamples.

Pointwise multiplication by the characteristic function of the half-space on anisotropic vector-valued function spaces

We study the pointwise multiplier property of the characteristic function of the half-space on weighted mixed-norm anisotropic vector-valued function spaces of Bessel potential and Triebel–Lizorkin type.