OPINION article Front. Med., 23 March 2023Sec. Infectious Diseases: Pathogenesis and Therapy Volume 10 - 2023 | https://doi.org/10.3389/fmed.2023.1126945

We focus on two problems relating to the question of when the product of two posinormal operators is posinormal, giving (1) necessary conditions and sufficient conditions for posinormal operators to have closed range, and (2) sufficient conditions for the product of commuting closed-range posinormal operators to be posinormal with closed range. We also discuss the relationship between posinormal operators and EP operators (as well as hypo-EP operators), concluding with a new proof of the Hartwig–Katz Theorem, which characterizes when the product of posinormal operators on Cn is posinormal.

A bounded linear operator A on a Hilbert space is posinormal if there exists a positive operator P such that AA⁎=A⁎PA. Posinormality of A is equivalent to the inclusion of the range of A in the range of its adjoint A⁎. Every hyponormal operator is posinormal, as is every invertible operator. We characterize both the posinormal and coposinormal composition operators Cφ on the Hardy space H2 of the open unit disk D when φ is a linear-fractional selfmap of D. Our work reveals that there are composition operators that are both posinormal and coposinormal yet have powers that fail to be posinormal.

We describe several shortcomings of a study by Patone et al, whose findings were recently published in the American Heart Association Journal Circulation, including the following: * The study's principal conclusion, as initially stated, begins "Overall, the risk of myocarditis is greater after SARS-CoV-2 infection than after COVID-19 vaccination ...." However, Patone et al never attempt to assess the incidence of myocarditis in their study population following SARS-CoV-2 infection. Rather, they make an untenable assumption that all infections occurring in their study population are associated with (reported) positive COVID-19 tests. Using publicly available data from the UK's ONS and NHS, we show that Patone et al's estimates, for the unvaccinated, of myocarditis incidence associated with infection are likely overestimated by a factor of at least 1.58. * The method Patone et al use to compute the incidence of myocarditis among the unvaccinated after a positive COVID test may overestimate risk. The authors assume, without justification, that unvaccinated persons hospitalized during the study period with positive-test-associated myocarditis would later choose to vaccinate with the same probability as unvaccinated persons who have had a positive COVID test. We present a plausibility argument that suggests a possible further exaggeration of myocarditis risk post infection by a factor of 1.5. * Patone et al fail to discuss important limitations of their study with respect to guiding public health recommendations. For instance, an insignificant number of cases contributing to the study's findings were Omicron-variant cases. Thus, the study's estimates of myocarditis risk following infection do not speak to the risk following Omicron infection, which is recognized to be milder than that of previous variants.

A bounded linear operator A on a Hilbert space ℋ is posinormal if there exists a positive operator P such that AA^* = A^*PA. We show that if A is posinormal with closed range, then A^n is posinormal and has closed range for all integers n≥ 1. Because the collection of posinormal operators includes all hyponormal operators, we obtain as a corollary that powers of closed-range hyponormal operators continue to have closed range. We also present a simple example of a closed-range operator T: ℋ→ℋ such that T^2 does not have closed range.

Let H^2(D) denote the classical Hardy space of the open unit disk D in the complex plane. We obtain descriptions of both the spectrum and essential spectrum of composition operators on H^2(D) whose symbols belong to the class S(2) introduced by Kriete and Moorhouse [Trans. Amer. Math. Soc., 359, 2007]. Our work reveals new possibilities for the shapes of composition-operator spectra, settling a conjecture of Cowen's [J. Operator Th. 9, 1983]. Our results depend on a number of lemmas, perhaps of independent interest, that provide spectral characterizations of sums of elements of a unital algebra over a field when certain pairwise products of the summands are zero.

A bounded operator T on a separable Hilbert space H is said to be complex symmetric if there exists an orthonormal basis for H with respect to which T has a self-transpose matrix representation. In this paper, we study the complex symmetry of composition operators Cϕf=f∘ϕ induced on the Hardy space H2 by holomorphic self-maps ϕ of the open unit disk D. For any holomorphic self-map ϕ of D, we establish that if Cϕ is complex symmetric, then ϕ must fix a point in D. Thus among the automorphisms of D, only the elliptic ones may induce complex symmetric composition operators. For an elliptic automorphism ϕ, we prove that if ϕ is not a rotation or of order 3, then Cϕ is complex symmetric if and only ifϕ(z)=α−z1−α¯z for some α∈D∖{0}.

Let X be a set of analytic functions on the open unit disk D, and let phi be an analytic function on D such that phi(D) is contained in D and f |-> f o phi takes X into itself. We present conditions on X ensuring that if f |-> f o phi is invertible on X, then phi is an automorphism of D, and we derive a similar result for mappings of the form f |-> psi.(f o phi), where psi is some analytic function on D. We obtain as corollaries of this purely function-theoretic work, new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as S^p and the weighted Hardy spaces H^2(beta).

We characterize those generating functions k that produce weighted Hardy spaces of the unit disk D supporting nontrivial Hermitian weighted composition operators. Our characterization shows that the spaces associated with the "classical reproducing kernels," as well as certain natural extensions of these spaces, are precisely those that are hospitable to Hermitian weighted composition operators. It also leads to a refinement of a necessary condition for a weighted composition to be Hermitian, obtained recently by Cowen, Gunatillake, and Ko, into one that is both necessary and sufficient.

We characterize the essentially normal weighted composition operators Cψ,φ on the Hardy space H 2 , whenever φ is a linear-fractional transformation and ψ ∈ A( ) . Also we investigate the essential normality problem for some other weighted composition operators on H .

We characterize the spectrum and essential spectrum of "essentially linear fractional" composition operators acting on the Hardy space H-two of the open unit disc U. When the symbols of these composition operators have Denjoy-Wolff point on the unit circle, the spectrum and essential spectrum coincide. Our work permits us to describe the spectrum and essential spectrum of certain associated weighted composition operators on the Hardy space.

We present a formal quantum mechanical analysis of the communication protocol of Prevedel {\it et al.}\ [Phys. Rev. Lett. \textbf{106}, 110505 (2011)], in which entanglement shared by sender and receiver is used to enhance, beyond that achievable via the optimal classical strategy, the probability of successful transmission of a bit through a particular noisy classical channel ${\cal N}$. We provide a full analysis of this protocol when the shared entanglement resides in a two-qubit system. Our analysis shows the measurement choices specified by the protocol yield the maximum possible enhancement of the probability of successful communication of the bit with one use of the channel ${\cal N}$. We determine that shared entanglement residing in a two-qudit system with $d > 2$ cannot provide enhancement beyond that produced by entangled qubits. Finally, we show how the protocol should be extended when probability parameters for the channel ${\cal N}$ are allowed to vary.

Let X be either the unit interval in ℝ or the unit disk \(\mathbb{D}\) in ℂ. Chalendar, Flattot, and Partington [2] study weighted composition operators Tw,ϒ on L2(X), where w ∈ L∞(X), ϒ : X → X is injective, and Tw,ϒf = wfoϒ for f ∈ L2(X). They introduce a (strict) partial order ≻ on X associated with Tw,ϒ and use it to obtain a sufficient condition for convergence of the sequence (Tn w,ϒyn) where (yn) is a backward minimal-vector sequence for Tw,ϒ. For the L2(\(\mathbb{D}\)) case, they give a detailed analysis of the situation where ϒ is linear-fractional. Through further study of the partial order ≻, we are able to generalize results from [2] that apply when ϒ is linear-fractional, replacing the linear-fractional hypotheses with univalence. In particular, our work yields generalizations of an invariant-subspace theorem in [2].

Under two-party deterministic dense-coding, Alice communicates (perfectly distinguishable) messages to Bob via a qudit from a pair of entangled qudits in pure state |Psi>. If |Psi> represents a maximally entangled state (i.e., each of its Schmidt coefficients is sqrt(1/d)), then Alice can convey to Bob one of d^2 distinct messages. If |Psi> is not maximally entangled, then Ji et al. [Phys. Rev. A 73, 034307 (2006)] have shown that under the original deterministic dense-coding protocol, in which messages are encoded by unitary operations performed on Alice's qudit, it is impossible to encode d^2-1 messages. Encoding d^2-2 is possible; see, e.g., the numerical studies by Mozes et al. [Phys. Rev. A 71, 012311 (2005)]. Answering a question raised by Wu et al. [Phys. Rev. A 73, 042311 (2006)], we show that when |Psi> is not maximally entangled, the communications limit of d^2-2 messages persists even when the requirement that Alice encode by unitary operations on her qudit is weakened to allow encoding by more general quantum operators. We then describe a dense-coding protocol that can overcome this limitation with high probability, assuming the largest Schmidt coefficient of |Psi> is sufficiently close to sqrt(1/d). In this protocol, d^2-2 of the messages are encoded via unitary operations on Alice's qudit, and the final (d^2-1)-th message is encoded via a (non-trace-preserving) quantum operation.

Let φ be an analytic function on the open unit disc U such that φ(U)⊆U, and let ψ be an analytic function on U such that the weighted composition operator Wψ,φ defined by Wψ,φf=ψf○φ is bounded on the Hardy space H2(U). We characterize those weighted composition operators on H2(U) that are unitary, showing that in contrast to the unweighted case (ψ≡1), every automorphism of U induces a unitary weighted composition operator. A conjugation argument, using these unitary operators, allows us to describe all normal weighted composition operators on H2(U) for which the inducing map φ fixes a point in U. This description shows both ψ and φ must be linear fractional in order for Wψ,φ to be normal (assuming φ fixes a point in U). In general, we show that if Wψ,φ is normal on H2(U) and ψ≢0, then φ must be either univalent on U or constant. Descriptions of spectra are provided for the operator Wψ,φ:H2(U)→H2(U) when it is unitary or when it is normal and φ fixes a point in U.

Let D be the open unit disc, let nu : D -> (0, infinity) be a typical weight, and let H(nu)(infinity) be the corresponding weighted Banach space consisting of analytic functions f on D such that parallel to f parallel to(nu):=sup(z is an element of D) nu(z)vertical bar f (z)vertical bar < infinity. We call H(nu)(infinity) a typical-growth space. For phi a holomorphic self-map of D, let C(phi) denote the composition operator induced by phi. We say that C(phi) is a bellwether for boundedness of composition operators on typical-growth spaces if for each typical weight nu, C(phi) acts boundedly on H(nu)(infinity) only if all composition operators act boundedly on H(nu)(infinity). We show that a sufficient condition for C(phi) to be a bellwether for boundedness is that phi have an angular derivative of modulus less than I at a point on partial derivative D. We raise the question of whether this angular-derivative condition is also necessary for C(phi) to be a bellwether for boundedness.

A useful method for deriving analytical results applicable to the standard two-party deterministic dense-coding protocol is introduced and illustrated. In this protocol, communication of K perfectly distinguishable messages is attainable via K selected local unitary operations performed on one qudit from a pair of entangled qudits of equal dimension d in a pure state parallel to psi > with largest Schmidt coefficient root lambda(0)>. The method takes advantage of the fact that the K message states, together with d(2)-K augmenting orthonormal state vectors, yield a unitary matrix, thereby implying properties of the K message states which otherwise are not readily recognized. Employing this augmented message matrix, we produce simple proofs of previously established results including (i) lambda(0)<= d/K, (ii) lambda(0)< d/K when K=d+1, and (iii) the impossibility of finding a parallel to psi > that can enable transmission of K=d(2)-1 messages but not d(2). Additional results obtained using the method include proofs that the lambda(0)<= d/K bound is reduced to at least (i) lambda(0)<=(1/2)[1+root(d-2)/(d+2)root] when K=d+1 and (ii) lambda(0)<=(K-m)/(2K-m-d) whenever (d+1)<= K <= 2d and the selected local unitaries include the first m non-negative integral powers of the shift operator X.

We present an analytical study of the standard two-party deterministic dense-coding protocol, under which communication of perfectly distinguishable messages takes place via a qudit from a pair of nonmaximally entangled qudits in a pure state vertical bar psi >. Our results include the following: (i) We prove that it is possible for a state vertical bar psi > with lower entanglement entropy to support the sending of a greater number of perfectly distinguishable messages than one with higher entanglement entropy, confirming a result suggested via numerical analysis in Mozes [Phys. Rev. A 71, 012311 (2005)]. (ii) By explicit construction of families of local unitary operators, we verify, for dimensions d=3 and d=4, a conjecture of Mozes about the minimum entanglement entropy that supports the sending of d+j messages, 2 <= j <= d-1; moreover, we show that the j=2 and j=d-1 cases of the conjecture are valid in all dimensions. (iii) Given that vertical bar psi > allows the sending of K messages and has root lambda(0) as its largest Schmidt coefficient, we show that the inequality lambda(0)<= d/K, established by Wu [Phys. Rev. A 73, 042311 (2006)], must actually take the form lambda(0)< d/K if K=d+1, while our constructions of local unitaries show that equality can be realized if K=d+2 or K=2d-1.

We study the intertwining relations between analytic Toeplitz operators induced on the Hardy space H^2 by analytic functions bounded on the open unit disc. Our work centers on the connection between intertwining between the Toeplitz operators the image containment between their symbols, as well as on the nature of the intertwining operator. We use our results to study the "extended eigenvalues" of analytic Toeplitz operators, i.e., the special case where the operator is intertwined with a scalar multiple of itself.