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Jani Virtanen

Publications and source records attributed to Jani Virtanen.

16 recordsLinked to original sources
arXiv2026

Sharp exponential integrability of conjugate functions

We prove that if a real-valued function $f\in L^1$ on the complex unit circle has a gap of width at least $π$ in its essential range, then $\exp(\tilde f)$ is not integrable, where $\tilde f$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ satisfying $$ \int_0^\infty Y(x) e^{-x}\, dx = \infty. $$ We also establish a local version on arcs of the unit circle: Under a natural condition on the inverse images of the two sides of the gap, $Y(\tilde f)$ fails to be integrable on the arc; in particular, it suffices that one of these inverse images is not an interval modulo null sets. Furthermore, we obtain corresponding results for complex-valued functions and a $Y$-integrability version of Zygmund's result.

math.CV↗
arXiv2026

Dilation Theory for Scaled $q$-Numerical Ranges

We generalise Berger's dilation theorem from numerical ranges to scaled $q$-numerical ranges. Specifically, we prove that if the scaled $q$-numerical range $\frac{W_q(A)}{q}$ is contained in the closed unit disk, then $A$ admits a unitary $ρ_q$-dilation, where $$ ρ_q=\max\left\{1,\frac{2|q|}{1+\sqrt{1-|q|^2}}\right\}. $$ As a corollary we show that every disk containing $Ω_q(A)$ is a complete $ρ_q$-spectral set for $A$. At $q=1$, this recovers the classical fact that any disk containing $W(A)$ is a complete $2$-spectral set for $A$.

math.FA↗
arXiv2026

The Complete Crouzeix Conjecture in Dimension Three and the Clouâtre-Ostermann-Ransford conjecture

We settle the complete Crouzeix conjecture for matrices of order at most three and prove the Q-algebra case of the completely bounded Clouâtre--Ostermann--Ransford (COR) conjecture for homomorphisms into matrices of order at most three, via sharp abstract column and row estimates. Our approach also establishes the scalar COR conjecture in a stronger form, for homomorphisms with commutative range on Banach algebras with unity satisfying von Neumann's inequality. Under contractivity of the symmetrized map, this result holds on arbitrary Hilbert spaces without initial boundedness assumptions on the homomorphism or the antilinear map. For arbitrary operator algebras, we disprove the complete COR conjecture by an exact three-dimensional example with target matrix order two. We also prove the sharp complete bound for every matrix subalgebra containing the diagonal, in arbitrary matrix order and on arbitrary target Hilbert spaces. We also obtain column and row square-function inequalities with sharp norm bounds, strict scalar bounds for operators similar to normal operators, sharp complete spectral constants for scaled $q$-numerical ranges in dimensions two and three, and rigidity, stability, and representing-measure results.

math.CV↗
arXiv2026

On the abstract approach to spectral constants: a proof of the Clouâtre--Ostermann--Ransford conjecture

Clouâtre, Ostermann, and Ransford formulated an abstract version of Crouzeix's conjecture involving a bounded unital homomorphism from a uniform algebra into matrices and a unital antilinear map. They conjectured that contractivity of the associated symmetrised map forces the homomorphism to have norm at most two. We prove this conjecture, in fact without requiring the antilinear map to be contractive and for homomorphisms into the bounded operators on a Hilbert space. The proof combines positivity of real parts, an operator-valued Herglotz theorem, and the perturbation lemma of Lorist and Schwenninger used in the recent proof of Crouzeix's conjecture.

math.FA↗
arXiv2026

A Counterexample to the Gau--Wang--Wu Conjecture on Partial Isometries

We disprove the conjecture of Gau, Wang and Wu that the numerical range of a finite-dimensional partial isometry, when circular, must be centred at the origin. More precisely, we prove that there is an \(\varepsilon>0\) such that, for every \(a\in(0,\varepsilon)\), one can find a rank-four partial isometry \(V_a\in\M_6(\R)\) satisfying $$ W(V_a)=\{ζ\in\C:|ζ-a|\leq 3/4\}. $$

math.FA↗
arXiv2026

Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces

Let $H$ be a complex Hilbert space and let $Ω\subset H$ be a domain. In infinite dimensions, there is no canonical complex Monge--Ampère operator and no basis-free determinant of the Levi form. Hence, a determinant-type characterization of maximal plurisubharmonic functions is not immediate. We propose to use the normalized determinants of Fujii and Seo: for a bounded strictly positive operator $A$ and a unit vector $x\in H$, we set $Δ_x(A):=\exp\bigl(\langle (\log A)x,x\rangle\bigr)$, and we extend this naturally to non-invertible positive operators. We show that, for strictly positive operators, inequalities for $Δ_x$ precisely describe the chaotic order $\log A\ge \log B$, and we combine this observation with Kantorovich--Specht type bounds for positive operators. For $u\in \mathcal{PSH}(Ω)\cap C^2(Ω)$ we define the \emph{Fujii--Seo determinant density} \[ \operatorname{FSD}(u)(a):=\inf_{\|x\|=1}Δ_x\!\bigl(D'D''u(a)\bigr),\qquad a\inΩ, \] and identify it with the lower spectral endpoint $\infσ(D'D''u(a))$. Thus, $\operatorname{FSD}(u)$ is precisely the infimum of the spectrum of the Levi form, and its vanishing gives a basis-independent criterion for pointwise degeneracy of the Levi form. We prove that maximality implies $\operatorname{FSD}(u)\equiv 0$, give sufficient global degeneracy criteria for maximality, and establish several comparison principles for $C^2$ plurisubharmonic functions, including results under uniform ellipticity bounds on the Levi form.

math.CV↗
arXiv2025

Toeplitz operators on large vector-valued Fock spaces

We characterize boundedness and compactness of Toeplitz operators on large vector-valued Fock spaces with Dall'Ara's weights [Adv.\ Math., 285 (2015) 1706--1740] in terms of generalized Berezin transforms, averaging functions, and Carleson measures. To determine Schatten class Toeplitz operators, we introduce the operator-valued Berezin transform and averaging functions.

math.FA↗
arXiv2023

Crouzeix's conjecture for classes of matrices

For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of $3 \times 3$ matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators.

math.FA↗
arXiv2021

Bounded and compact Toeplitz+Hankel matrices

We show that an infinite Toeplitz+Hankel matrix $T(φ) + H(ψ)$ generates a bounded (compact) operator on $\ell^p(\mathbb{N}_0)$ with $1\leq p\leq \infty$ if and only if both $T(φ)$ and $H(ψ)$ are bounded (compact). We also give analogous characterizations for Toeplitz+Hankel operators acting on the reflexive Hardy spaces. In both cases, we provide an intrinsic characterization of bounded operators of Toeplitz+Hankel form similar to the Brown-Halmos theorem. In addition, we establish estimates for the norm and the essential norm of such operators.

math.FA↗
arXiv2020

Compact Hankel Operators with Bounded Symbols

We discuss the compactness of Hankel operators on Hardy, Bergman and Fock spaces with focus on the differences between the three cases, and complete the theory of compact Hankel operators with bounded symbols on the latter two spaces with standard weights. In particular, we give a new proof (using limit operator techniques) of the result that the Hankel operator $H_f$ is compact on Fock spaces if and only if $H_{\bar f}$ is compact. Our proof fully explains that this striking result is caused by the lack of nonconstant bounded analytic functions in the complex plane (unlike in the other two spaces) and extends the result from the Fock-Hilbert space to all Fock-Banach spaces. As in Hardy spaces, we also show that the compactness of Hankel operators is independent of the underlying space in the other two cases.

math.FA↗
arXiv2017

On generalized Toeplitz and little Hankel operators on Bergman spaces

We find a concrete integral formula for the class of generalized Toeplitz operators $T_a$ in Bergman spaces $A^p$, $1<p<\infty$, studied in an earlier work by the authors. The result is extended to little Hankel operators. We give an example of an $L^2$-symbol $a$ such that $T_{|a|} $ fails to be bounded in $A^2$, although $T_a : A^2 \to A^2$ is seen to be bounded by using the generalized definition. We also confirm that the generalized definition coincides with the classical one whenever the latter makes sense.

math.FA↗
arXiv2014

Study of Negative and Positive Superhumps in ER Ursae Majoris

We carried out the photometric observations of the SU UMa-type dwarf nova ER UMa during 2011 and 2012, which showed the existence of persistent negative superhumps even during the superoutburst. We performed two-dimensional period analysis of its light curves by using a method called "least absolute shrinkage and selection operator" (Lasso) and "phase dispersion minimization" (PDM) analysis, and we found that the period of negative superhumps systematically changed between a superoutburst and the next superoutburst. The trend of the period change can beinterpreted as reflecting the change of the disk radius. This change of the disk radius is in good agreement with the predicted change of the disk radius by the thermal-tidal instability (TTI) model. The normal outbursts within a supercycle showed a general trend that the rising rate to maximum becomes slower as the next superoutburst approaches. The change can be interpreted as the consequence of the increased gas-stream flow onto the inner region of the disk as the result of the tilted disk. Some of the superoutbursts were found to be triggered by a precursor normal outburst when the positive superhumps appeared to develop. The positive and negative superhumps co-existed during the superoutburst. The positive superhumps were prominent only during four or five days after the supermaximum, while the signal of the negative superhumps became strong after the middle phase of the superoutburst plateau. A simple combination of the positive and negative superhumps was found to be insufficient in reproducing the complex profile variation. We were able to detect the developing phase of positive superhumps (stage A superhumps) for the first time in ER UMa-type dwarf novae. Using the period of stage A superhumps, we obtained a mass ratio of 0.100(15), which indicates that ER UMa is on the ordinary evolutional track of CVs.

astro-ph.SR↗
arXiv2012

Discovery of Negative Superhumps during a Superoutburst of January 2011 in ER Ursae Majoris

We report on a discovery of "negative" superhumps during the 2011 January superoutburst of ER UMa. During the superoutburst which started on 2011 January 16, we detected negative superhumps having a period of 0.062242(9) d, shorter than the orbital period by 2.2%. No evidence of positive superhumps was detected during this observation. This finding indicates that the disk exhibited retrograde precession during this superoutburst, contrary to all other known cases of superoutbursts. The duration of this superoutburst was shorter than those of ordinary superoutbursts and the intervals of normal outbursts were longer than ordinary ones. We suggest a possibility that such unusual outburst properties are likely a result of the disk tilt, which is supposed to be a cause of negative superhumps: the tilted disk could prevent the disk from being filled with materials in the outmost region which is supposed to be responsible for long-duration superoutbursts in ER UMa-type dwarf novae. The discovery signifies the importance of the classical prograde precession in sustaining long-duration superoutbursts. Furthermore, the presence of pronounced negative superhumps in this system with a high mass-transfer rate favors the hypothesis that hydrodynamical lift is the cause of the disk tilt.

astro-ph.SR↗
arXiv2012

Survey of Period Variations of Superhumps in SU UMa-Type Dwarf Novae. III: The Third Year (2010--2011)

Continuing the project described by Kato et al. (2009, PASJ 61, S395, arXiv:0905.1757), we collected times of superhump maxima for 51 SU UMa-type dwarf novae mainly observed during the 2010-2011 season. Although most of the new data for systems with short superhump periods basically confirmed the findings by Kato et al. (2009) and Kato et al. (2010, PASJ 62, 1525, arXiv:1009.5444), the long-period system GX Cas showed an exceptionally large positive period derivative. An analysis of public Kepler data of V344 Lyr and V1504 Cyg yielded less striking stage transitions. In V344 Lyr, there was prominent secondary component growing during the late stage of superoutbursts, and the component persisted at least for two more cycles of successive normal outbursts. We also investigated the superoutbursts of two conspicuous eclipsing objects: HT Cas and the WZ Sge-type object SDSS J080434.20+510349.2. Strong beat phenomena were detected in both objects, and late-stage superhumps in the latter object had an almost constant luminosity during the repeated rebrightenings. The WZ Sge-type object SDSS J133941.11+484727.5 showed a phase reversal around the rapid fading from the superoutburst. The object showed a prominent beat phenomenon even after the end of the superoutburst. A pilot study of superhump amplitudes indicated that the amplitudes of superhumps are strongly correlated with orbital periods, and the dependence on the inclination is weak in systems with inclinations smaller than 80 deg.

astro-ph.SR↗
arXiv2006

Norms of Toeplitz Matrices with Fisher-Hartwig Symbols

We describe the asymptotics of the spectral norm of finite Toeplitz matrices generated by functions with Fisher-Hartwig singularities as the matrix dimension goes to infinity. In the case of positive generating functions, our result provides the asymptotics of the largest eigenvalue, which is of interest in time series with long-range memory.

math.FA↗