This paper investigates isometries on orthogonal direct sums of Kreĭn spaces and subspaces. Internal sums (i-sums) of pairwise orthogonal regular subspaces 𝔎_n , n≥ 0 of a Kreĭn space 𝔎 associated with given fundamental symmetries J_n , n≥ 0 are introduced as subspaces isomorphic with external sums. A characterization of multiplication operators between external and internal sums is provided in terms of the uniform norm boundedness of the restrictions to the components. We introduce internal shifts (i-shifts) as operators unitarily equivalent to standard shifts on Hardy-type Kreĭn spaces and establish several equivalent characterizations. A Wold-type decomposition theorem for isometries on Kreĭn spaces is proved, decomposing an isometry into a unitary part and an i-shift under suitable regularity conditions. We study (minimal) unitary extensions U on a Kreĭn space 𝔎 of bounded isometries acting on a Kreĭn subspace ℌ . The regularity of the minimal subspace ⋁ _n≤ 0U^nℌ is characterized in terms of the convergence of (U^*nPU^n)_n≥ 0 , where P is the orthogonal projection onto ℌ . Conditions for U^*|_𝔎⊖ℌ to be a shift or an i-shift are established. We study the possibility that shifts on Kreĭn spaces have minimal unitary extensions of bilateral shift type. We prove, for example, that V and U^*|_𝔎⊖ℌ are i-shifts if and only if each k∈𝔎 can be represented as k=∑ _n=-∞^∞ U^nr_n , where the sequence r_n=(U^*PU-P)U^*nk , n≥ 0 of the Fourier coefficients is norm square summable. Finally, necessary and sufficient conditions for the uniqueness of unitary extensions are given in terms of the nature of V^*.
We extend the study of normal projections in Krein spaces to the unbounded case. We characterize both weakly normal and normal projections, and prove that every normal projection admits a decomposition as the sum of a selfadjoint projection and a closed projection with neutral range. We show that every closed subspace 𝒮 is the range of a (possibly unbounded) normal projection and parametrize the set of normal projections onto 𝒮 using the notion of normal companions.
We introduce and study Li-Yorke chaos for sequences of continuous linear operators from an F -space to a normed space. We show that in every infinite-dimensional separable complex Banach space, there exists a sequence of operators with a dense set of irregular vectors but without a dense irregular manifold. We introduce the D -phenomenon to establish a common dense lineability criterion that encompasses properties such as recurrence, universality, and Li-Yorke chaos.
Let G be an infinite, compact abelian group, E be a homogeneous Banach space over G and ℒ( E) be the space of all continuous linear operators from E into itself equipped with the operator norm. Translation operators are isometries in E (by definition) and so the closed subalgebra 𝔪( E) of ℒ( E) consisting of those operators which commute with all translations is well defined. It is shown that there exists a contractive projection 𝒬 of ℒ( E) onto 𝔪( E) which is positivity preserving. Moreover, every operator 𝒬( T) ∈𝔪( E) , with T∈ℒ( E) , is induced by a unique Fourier multiplier function T̂∈ℓ ^∞( Γ) , where Γ is the dual group of G. In the setting of the homogeneous Banach spaces L^p( G) , for 1≤ p<∞ and G an amenable group, these results are due to W. Arendt and J. Voigt.
Let $$\mathscr {H}^\infty $$ H ∞ be the set of all Dirichlet series $$\textstyle f\!=\!{{\sum \limits _{n=1}^\infty }} a_nn^{-s}$$ f = ∑ n = 1 ∞ a n n - s (where $$a_n\!\in \mathbb {C}$$ a n ∈ C for all $$n\!\in \! \mathbb {N}\!=\!\{1,2,3,\cdots \}$$ n ∈ N = { 1 , 2 , 3 , ⋯ } ) that converge at each s in the half-plane $$\mathbb {C}_0\!:=\!\{s\!\in \! \mathbb {C}\!:\! \text {Re}(s)\!>\!0\}$$ C 0 : = { s ∈ C : Re ( s ) > 0 } , such that $$\Vert f\Vert _{\infty }\!=\!\sup _{s\in \mathbb {C}_0}\!|f(s)|\!<\!\infty $$ ‖ f ‖ ∞ = sup s ∈ C 0 | f ( s ) | < ∞ . Then $$\mathscr {H}^\infty $$ H ∞ is a Banach algebra with pointwise operations and the supremum norm $$\Vert \cdot \Vert _\infty $$ ‖ · ‖ ∞ , and has been studied in earlier works. The article introduces a new family of Banach subalgebras $$\mathscr {H}^\infty _{S}$$ H S ∞ of $$\mathscr {H}^\infty $$ H ∞ . For $$S\!\subset \! \mathbb {N}$$ S ⊂ N , let $$\mathscr {H}^\infty _{S}$$ H S ∞ be the set of all elements $$\textstyle {{\sum \limits _{n=1}^\infty }} a_nn^{-s}\in \mathscr {H}^\infty $$ ∑ n = 1 ∞ a n n - s ∈ H ∞ such that for all $$n\in \mathbb {N}\setminus S$$ n ∈ N \ S , $$a_n\!=\!0$$ a n = 0 . Then $$\mathscr {H}^\infty _{S}$$ H S ∞ is a unital Banach subalgebra of $$\mathscr {H}^\infty $$ H ∞ with the $$\Vert \cdot \Vert _\infty $$ ‖ · ‖ ∞ norm if and only if S is a multiplicative subsemigroup of $$\mathbb {N}$$ N containing 1. It is shown that for such S , $$\mathscr {H}^\infty _{S}$$ H S ∞ is the multiplier algebra of $$\mathscr {H}^2_S$$ H S 2 , where $$\mathscr {H}^2_S$$ H S 2 is the Hilbert space of all $$ \textstyle f\!=\!{{\sum \limits _{n\in S}}} a_nn^{-s}$$ f = ∑ n ∈ S a n n - s such that $$\Vert f\Vert _2\!:=\!({{\sum \limits _{n\in S}}} |a_n|^2)^{\frac{1}{2}}\!<\!\infty $$ ‖ f ‖ 2 : = ( ∑ n ∈ S | a n | 2 ) 1 2 < ∞ . A characterisation of the group of units in $$\mathscr {H}^\infty _{S}$$ H S ∞ is given, by showing an analogue of the Wiener 1/ f theorem for $$\mathscr {H}^\infty _{S}$$ H S ∞ . If S has an infinite set of generators allowing a unique representation of each element of S , then it is shown that the Bass stable rank of $$\mathscr {H}^\infty _S$$ H S ∞ is infinite.
We introduce and rigorously analyze two distinct classes of Hardy-type operators on the p-adic vector space ℚ_p^n . The first class consists of linear operators {ℋ_t}_t>0 acting on L^q(ℚ_p^n) , 1 ≤ q < ∞ , forming a strongly continuous, translation-invariant, sub-Markovian semigroup. We establish boundedness, regularity properties, and invariance on Bruhat-Schwartz test functions. We develop a rigorous probabilistic interpretation, proving that these operators define transition kernels of a spatially homogeneous Lévy-type Markov process on ℚ_p^n with pure jump dynamics. Numerical experiments validate exponential convergence to steady state and confirm the spectral gap. The second class comprises nonlinear operators {ℋ_t}_t>0 on C_0(ℚ_p^n) , proven to be well-defined contractions satisfying the positive maximum principle. We establish m-dissipativity, guaranteeing the generation of a strongly continuous nonlinear semigroup. Computational examples illustrate spatial confinement and amplitude decay characteristic of dissipative ultrametric dynamics. This framework establishes rigorous connections between Hardy operator theory and stochastic processes on ultrametric spaces.
Let X be a Banach function space over the unit circle 𝕋 , let X' be its associate space, and let H[X] and H[X'] be abstract Hardy spaces built upon X and X' , respectively. Suppose the Riesz projection P is bounded from X onto H[X]. We say that a function a invertible in L^∞ admits a Wiener-Hopf factorisation in X if it can be written as a=a_-e_κ a_+ , where the middle factor is e_κ (t)=t^κ with t∈𝕋 and κ∈ℤ , the outmost factors satisfy a_-∈ H[X], a_-^-1∈ H[X'], a_+∈ H[X'], a_+^-1∈ H[X], and the operator P is bounded on the weighted space X(|a_+^-1|) . We prove that a Toeplitz operator T(a)f:=P(af) with symbol a∈ L^∞ is Fredholm on H[X] if and only if a is invertible in L^∞ and admits a Wiener-Hopf factorisation in X. In this case, the Fredholm index of T(a) is equal to -κ . No assumptions on separability or reflexivity of X are made, which significantly complicates the matters. Our result extends a scalar version of Simonenko’s factorisation theorem obtained in the 1960s for Lebesgue spaces.
An operator A on a complex Hilbert space ℋ is called antilinear if A(x+y)=Ax+Ay and A(λ x)=λ Ax for x,y∈𝒟(A) and λ∈ℂ . We investigate some classes of densely defined antilinear unbounded operators, especially antilinear normal operators. We give various characterizations of antilinear normal operators and study a class of such operators in detail. Our main result is a structure theorem for unbounded antilinear normal operators.
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.
The set ofL-resolvents of a densely defined symmetric opera-tor in a Hilbert spaceHwith a proper gauge L(subset of H) was described byKrein and Saakyan. The Krein-Saakyan theory ofL-resolvent matriceswas extended by Shmul'yan and Tsekanovskii to the case of impropergaugeL(L subset of H) and by Langer and Textorius to the case of symmetriclinear relations in Hilbert spaces. In the present paper we find connec-tions between the theory of boundary triples and the Krein-Saakyantheory of L-resolvent matrices for symmetric linear relations with im-proper gauges in Hilbert spaces and extend the known formula for theL-resolvent matrix in terms of boundary operators to this class of re-lations. Descriptions of spectral and pseudo-spectral functions of sym-metric linear relations with improper gauges are given. The results areapplied to linear relations generated by a canonical system.
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 as the adjoint operator. These results are then used to determine Fredholm characteristics, and to determine when the unbounded Toeplitz operators are symmetric and have a selfadjoint extension.
We obtain a simpler proof of a result in [J. Funct. Anal. 278 (2020), 108401]. By using an elimination method, we completely characterize the bounded and compact differences of two generalized weighted composition operators with the same order, i.e. C_u,φ^(n)-C_v,ψ^(n) , on the standard weighted Bergman spaces. Furthermore, we show a rigidity of the difference of two generalized weighted composition operators with different orders, i.e. C_u,φ^(n)-C_v,ψ^(m)(n m) . The boundedness, compactness and Hilbert-Schmidt nature of C_u,φ^(n)-C_v,ψ^(m) are equivalent to those of C_u,φ^(n)-C_v,ψ^(m) . And the compact difference of the sum, i.e. (C_u_1,φ^(n)+C_u_2,φ^(m))-(C_v_1,ψ^(n)+C_v_2,ψ^(m)) is also studied.
Let 𝔻 be the open unit disk in the complex plane ℂ and let ℳ be a semifinite von Neumann algebra. The main result of this paper is the weak type (1, 1) inequality of the weighted Bergman projection P_w induced by reproducing kernels K_z(ζ )=1/(1-z̅ζ )^γ∫ _0^1 d ν (r)/1-r z̅ζ , that is, if v∈ B_1,w , then: ‖ P_w(f)‖ _L^v_1, ∞(𝒩_𝔻)≤ CB_1,w(v)^2‖ f‖ _L^v_1(𝒩_𝔻), where 𝒩_𝔻=L_∞(𝔻,A_w)⊗̅ℳ , A_w is the normalized Lebesgue area measure related to weight w. Moreover, P_w is bounded on L^v_p (𝒩_𝔻) with 1
In analogy to normal eigenvalues we introduce approximately normal eigenvalues of unbounded operators in Banach spaces. Roughly speaking, those are isolated eigenvalues of the approximate point spectrum of closed range and finite algebraic multiplicity. Thereby we define an approximate version of the discrete spectrum and study its complement in the approximate point spectrum. In the bounded case, this turns out to be a well-known type of essential spectrum introduced in [39] for which we prove natural, yet, to the best knowledge of the author, new characterisations. We study the connexion of this type of essential spectrum with the essential numerical ranges in Hilbert spaces. As a final result we show that the essential numerical range contains the whole approximate point spectrum except, possibly, some isolated eigenvalues of the approximate point spectrum of finite multiplicity.
Using a generalized Birman–Schwinger principle developed in [31] for operators formally given by H_0 + V and the theory of Sobolev multipliers, we develop Birman–Schwinger principles for the following concrete situations: one-dimensional Schrödinger and massless relativistic Schrödinger operators with distributional potentials from H^-1(ℝ) and H^-(1/2)+δ(ℝ) for some δ∈ (0,1/2) , respectively; two-dimensional Schrödinger operators with distributional potentials in H^-1+δ(ℝ^2) for some δ∈ (0,1) ; and three-dimensional Schrödinger operators with distributional potentials in H^-1/2(ℝ^3) . In all cases, the Birman–Schwinger operator A_V(λ ) = - (H_0-λ I_L^2(ℝ^n) )^-1/2V (H_0-λ I_L^2(ℝ^n) )^-1/2, λ∈ (-∞ ,0) (here either H_0=-Δ with n∈{1,2,3} or H_0=D_0=(-Δ )^1/2 with n=1 ), is Hilbert–Schmidt with norm given explicitly by a weighted integral involving the Fourier transform of the distribution V.
Conditions are established for rank three partial isometries to have circular components contained in their Kippenhahn curves. In particular, such matrices with circular numerical ranges are described. It is also established that the Gau–Wang–Wu conjecture holds for matrices under consideration.
Let ℋ^∞ be the set of all Dirichlet series f=∑ _n=1^∞ a_nn^-s (where a_n∈ℂ for all n∈ℕ={1,2,3,⋯} ) that converge at each s in the half-plane ℂ_0:={s∈ℂ:Re(s)>0} , such that ‖ f‖ _∞=sup _s∈ℂ_0|f(s)|<∞ . Then ℋ^∞ is a Banach algebra with pointwise operations and the supremum norm ‖·‖ _∞ , and has been studied in earlier works. The article introduces a new family of Banach subalgebras ℋ^∞ _S of ℋ^∞ . For S⊂ℕ , let ℋ^∞ _S be the set of all elements ∑ _n=1^∞ a_nn^-s∈ℋ^∞ such that for all n∈ℕ∖ S , a_n=0 . Then ℋ^∞ _S is a unital Banach subalgebra of ℋ^∞ with the ‖·‖ _∞ norm if and only if S is a multiplicative subsemigroup of ℕ containing 1. It is shown that for such S, ℋ^∞ _S is the multiplier algebra of ℋ^2_S , where ℋ^2_S is the Hilbert space of all f=∑ _n∈ S a_nn^-s such that ‖ f‖ _2:=(∑ _n∈ S |a_n|^2)^1/2<∞ . A characterisation of the group of units in ℋ^∞ _S is given, by showing an analogue of the Wiener 1/f theorem for ℋ^∞ _S . If S has an infinite set of generators allowing a unique representation of each element of S, then it is shown that the Bass stable rank of ℋ^∞ _S is infinite.
We consider Maxwell’s equations with perfect electric conductor boundary conditions in three-dimensional unbounded domains which are the union of a bounded resonator and one or several semi-infinite waveguides. We are interested in the existence of electromagnetic trapped modes, i.e. L^2 solutions of the problem without source term. These trapped modes are associated to eigenvalues of Maxwell’s operator, that can be either below the essential spectrum or embedded in it. First for homogeneous waveguides, we present different families of geometries for which we can prove the existence of eigenvalues. Then we exhibit certain non-homogeneous waveguides with local perturbations of the dielectric constants that support trapped modes. Let us mention that some of the mechanisms we propose are very specific to Maxwell’s equations and have no equivalent for the scalar Dirichlet or Neumann Laplacians.
