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In this paper, we provide a version of the classical result of Kowalski and S\l odkowski that generalizes the famous Gleason-Kahane-$\dot{\rm Z}$elazko (GKZ) theorem by characterizing multiplicative linear functionals amongst all complex-valued functions on a Banach algebra. We first characterize maps on $\mathcal{A}$-valued polynomials of several variables that satisfy some conditions, motivated by the result of Kowalski and S\l odkowski, as a composition of a multiplicative linear functional on $\mathcal{A}$ and a point evaluation on the polynomials, where $\mathcal{A}$ is a complex Banach algebra with identity. We then apply it to prove an analogue of Kowalski and S\l odkowski's result on topological spaces of vector-valued functions of several variables. These results extend our previous work from \cite{jaikishan2024multiplicativity}; however, the techniques used differ from those used in \cite{jaikishan2024multiplicativity}. Furthermore, we characterize weighted composition operators between Hardy spaces over the polydisc amongst the continuous functions between them. Additionally, we register a partial but noteworthy success toward a multiplicative GKZ theorem for Hardy spaces.

This article describes Hilbert spaces contractively contained in certain reproducing kernel Hilbert spaces of analytic functions on the open unit disc which are nearly invariant under division by an inner function. We extend Hitt's theorem on nearly invariant subspaces of the backward shift operator on $H^2(\bb D)$ as well as its many generalizations to the setting of de Branges spaces.

This paper presents a fairly general version of the well-known Gleason-Kahane-$\dot{\text{Z}}$elazko (GKZ) theorem in the spirit of a GKZ type theorem obtained recently by Mashreghi and Ransford for Hardy spaces. In effect, we characterize a class of linear functionals as point evaluations on the vector space of all complex polynomials $\cl P$. We do not make any topological assumptions on $\cl P$. We then apply this characterization to present a version of the GKZ theorem for a vast class of topological spaces of complex-valued functions including the Hardy, Bergman, Dirichlet, and many more well-known function spaces. We obtain this result under the assumption of continuity of the linear functional, which we show, with the help of an example, to be a necessary condition for the desired conclusion. Lastly, we use the GKZ theorem for polynomials to obtain a version of the GKZ theorem for strictly cyclic weighted Hardy spaces.

We present stellar variability towards the young open cluster NGC\,6823. Time series $V$- and $I$-band CCD photometry led to identification and characterization of 88 variable stars, of which only 14 have been previously recognized. We ascertain the membership of each variable with optical $UBVI$ and infrared photometry, and with Gaia EDR3 parallax and proper motion data. Seventy two variable stars are found to be cluster members, of which 25 are main sequence stars and 48 are pre-main-sequence stars. The probable cluster members collectively suggest an isochrone age of the cluster to be about 2~Myrs based on the GAIA photometry. With the color and magnitude, as well as the shape of the light curve, we have classified the main sequence variables into $\beta$~Cep, $\delta$~Scuti, slowly pulsating B type, and new class variables. Among the pre-main-sequence variables, eight are classical T Tauri variables, and four are Herbig Ae/Be objects, whereas the remaining belong to the weak-lined T Tauri population. The variable nature of 32 stars is validated with TESS light curves. Our work provides refined classification of variability of pre-main-sequence and main-sequence cluster members of the active star-forming complex, Sharpless\,86. Despite no strong evidence of the disk-locking mechanism in the present sample of TTSs, one TTS with larger $\Delta(I-K)$ is found to be slow rotator., Comment: 19 pages, 19 figures, accepted for publication in MNRAS

In this paper we study subspaces which are invariant under squares and cubes (separately as well as jointly) of unicellular backward weighted shift operators on a separable Hilbert space. The finite-dimensional subspaces are characterized for all weights and the infinite-dimensional subspaces are characterized for two classes of weights.

The $UBVRI$ CCD photometric data of open star cluster NGC 1513 are obtained with the 3.6-m Indo-Belgian Devasthal optical telescope (DOT). Analyses of the GAIA EDR3 astrometric data have identified 106 possible cluster members. The mean proper motion of the cluster is estimated as $\mu_{\alpha}Cos{\delta}=1.29\pm0.02$ and $\mu_{\delta}=-3.74\pm0.02$ mas yr$^{-1}$. Estimated values of reddening $E(B-V)$ and distance to the NGC 1513 are 0.65$\pm$0.03 mag and 1.33$\pm$0.1 kpc respectively. An age of $225\pm25$ Myr is assigned to the cluster by comparing theoretical isochrones with deep observed cluster sequence. Using observations taken with the 3.6-m DOT, values of distance and age of the galactic globular cluster NGC 4147 are estimated as $18.2\pm0.2$ Kpc and $14\pm2$ Gyr respectively. The optical observations of planetary transit around white dwarf WD1145+017 and $K$-band imaging of star-forming region Sharpless Sh 2-61 demonstrate observing capability of 3.6-m DOT. Optical and near-infrared observations of celestial objects and events are being carried out routinely with the 3.6-m DOT. They indicate that the performance of the telescope is at par with those of other similar telescopes located elsewhere in the world. We therefore state that this observing facility augurs well for multi-wavelength astronomy including study of astrophysical jets., Comment: Accepted for Special Issue of Journal of Astrophysics and Astronomy, 2022, Astrophysical jets and observational facilities: National perspective, 05-09 April 2021, ARIES Nainital

This paper deals with representing in concrete fashion those Hilbert spaces that are vector subspaces of the Hardy spaces $H^p(\bb D^n) \ (1\le p\le \infty)$ that remain invariant under the action of coordinate wise multiplication by an $n$-tuple $(T_{B_1},\dots, T_{B_n})$ of operators where each $B_i, \ 1\le i\le n,$ is a finite Blaschke factor on the open unit disc. The critical point to be noted is that these $T_{B_i}$ are assumed to be weaker than isometries as operators. Thus our main theorem extends the principal result of \cite{LS} in the following three directions: $(i)$ from one to several variables; $(ii)$ from multiplication with the coordinate function $z$ to an $n$-tuple of multiplication by finite Blaschke factors $B_i, \ 1\le i\le n;$ $(iii)$ from vector subspaces of $H^2(\bb D)$ to the case of vector subspaces of $H^p(\bb D^n), \ 1\le p\le \infty.$ We further derive a generalization of Slocinski's well known Wold type decomposition of a pair of doubly commuting isometries to the case of $n$-tuple of doubly commuting operators whose actions are weaker than isometries.

In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr's inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n,\ n\ge 2$., Comment: Accepted in Proc. Amer. Math. Soc. (2021)

We identify 228 periodic variables in the region of young open cluster NGC 281 using time series photometry carried out from 1 m class ARIES telescopes, Nainital. The cluster membership of these identified variables is determined on the basis colour-magnitude, two colour diagrams and kinematic data. Eighty one variable stars are found to be members of the cluster NGC 281. Of 81 variables, 30 and 51 are probable main-sequence and pre-main-sequence members, respectively. Present study classifies main-sequence variable stars into different variability types according to their periods of variability, shape of light curves and location in the Hertzsprung-Russell diagram. These identified main-sequence variables could be $\beta$ Cep, $\delta$ Scuti, slowly pulsating B type and new class variables. Among 51 pre-main-sequence variables, majority of them are weak line T Tauri stars. The remaining 147 variables could belong to the field population. The variability characteristics of the field population indicate that these variables could be RR Lyrae, $\delta$ Scuti and binaries type variables., Comment: 17 pages, 17 figures, accepted for publication in MNRAS

We present here optical I-band photometric variability study down to $\simeq$ 19 mag of a young ($\sim$2-3 Myr) star-forming region IC 348 in the Perseus molecular cloud. We aim to explore the fast rotation (in the time-scales of hours) in Very Low Mass stars (VLMs) including Brown Dwarfs (BDs). From a sample of 177 light-curves using our new I-band observations, we detect new photometric variability in 22 young M-dwarfs including 6 BDs, which are bonafide members in IC 348 and well-characterized in the spectral type of M-dwarfs. Out of 22 variables, 11 M dwarfs including one BD show hour-scale periodic variability in the period range 3.5 - 11 hours and rest are aperiodic in nature. Interestingly, an optical flare is detected in a young M2.75 dwarf in one night data on 20 December 2016. From the flare light curve, we estimate the emitted flared energy of 1.48 $\times$ 10$^{35}$ ergs. The observed flared energy with an uncertainty of tens of per cent is close to the super-flare range ($\sim$ 10$^{34}$ ergs), which is rarely observed in active M dwarfs., Comment: 12 pages, 5 figures

We present here identification and characterization of the young stellar population associated with an active star-forming site Sh2-242. We used our own new optical imaging and spectroscopic observational data, as well as several archival catalogs, e.g., Pan-STARRS 1, $Gaia$ DR2, IPHAS, WIRCam, 2MASS, and $Spitzer$. Slit spectroscopic results confirm the classification of the main ionizing source BD+26 980 as an early-type star of spectral type B0.5 V. The spectrophotometric distance of the star is estimated as 2.08 $\pm$ 0.24 kpc, which confirms the source as a member of the cluster. An extinction map covering a large area (diameter $\sim$ 50') is generated with $H$ and $K$ photometry toward the region. From the map, three distinct locations of peak extinction complexes ($A_{V}$ $\simeq$ 7$-$17 mag) are identified for the very first time. Using the infrared color excess, a total of 33 Class I and 137 Class II young objects are classified within the region. The IPHAS photometry reveals classification of 36 H$\alpha$ emitting sources, which might be class II objects. Among 36 H$\alpha$ emitting sources, 5 are already identified using infrared excess emission. In total, 201 young objects are classified toward S242 from this study. The membership status of the young sources is further windowed with the inclusion of parallax from the $Gaia$ DR2 catalog. Using the optical and infrared color-magnitude diagrams, the young stellar objects are characterized with an average age of $\sim$ 1 Myr and the masses in the range 0.1$-$3.0 $M_\odot$. The census of the stellar content within the region is discussed using combined photometric and spectroscopic data., Comment: 17 pages, 12 figures, Published in the Astronomical Journal

We present time series photometry in the field of Stock 8 and identified 130 short period variable stars. Twenty eight main-sequence and 23 pre-main-sequence variables are found to be part of cluster Stock 8. The main-sequence variables are classified as slow pulsator of the B type, $\beta$ Cep and $\delta$ Scuti stars. Fourteen main-sequence stars could be new class variables as discussed by Mowlavi et al. (2013) and Lata et al. (2014). The age and mass of pre-main-sequence variables are found to be $\lesssim$ 5 Myr and in the mass range of 0.5 to 2.8 M$_{\odot}$, respectively. These pre-main-sequence stars could be T-Tauri variables. We have found 5 and 2 of 23 PMS variables as classical T-Tauri stars and Herbig Ae/Be stars, respectively, whereas 16 PMS stars are classified as weak-line T Tauri stars., Comment: 17 pages, 14 figures, Accepted for the publication in Astronomical Journal

We present results of a search for variable stars in a region of the globular cluster NGC 4147 based on photometric observations with 4k x 4k CCD imager mounted at the axial port of the recently installed 3.6 m Devasthal optical telescope at Aryabhatta Research Institute of Observational Sciences, Nainital, India. We performed time series photometry of NGC 4147 in V and R bands, and identified 42 periodic variables in the region of NGC 4147, 28 of which have been detected for the first time. Seventeen variable stars are located within the half light radius $\lesssim$ 0.48 arcmin, of which 10 stars are newly identified variables. Two of 10 variables are located within the core radius $\lesssim$ 0.09 arcmin. Based on the location in the $V/(V-R)$ colour magnitude diagram and variability characteristics, 7, 8, 5 and 1 newly identified probable member variables are classified as RRc, EA/E, EW and SX Phe, respectively. The metallicity of NGC 4147 estimated from light curves of RRab and RRc stars with the help of Fourier decomposition is found to be characteristics of Oosterhoff II. The distance derived using light curves of RRab stars is consistent with that obtained from the observed $V/(V-R)$ colour-magnitude diagram., Comment: 20 pages, 13 Figures, accepted for publication in Astronomical Journal

In this note we study sub-Hardy Hilbert spaces on which the the action of the operator of multiplication by the coordinate function z is assumed to be weaker than that of an isometry. We identify such operators with a class of weighted shifts. The well known results of de Branges and Beurling are deduced as corollaries ., Comment: 8 Pages

We present time series photometry of 57 variable stars in the cluster region NGC 7380. The association of these variable stars to the cluster NGC 7380 has been established on the basis of two colour diagrams and colour-magnitude diagrams. Seventeen stars are found to be main-sequence variables, which are mainly B type stars and are classified as slowly pulsating B stars, $\beta$ Cep or $\delta$ Scuti stars. Some of them may belong to new class variables as discussed by Mowlavi et al. (2013) and Lata et al. (2014). Present sample also contains 14 pre-main-sequence stars, whose ages and masses are found to be mostly $\lesssim$ 5 Myr and range 0.60 $\lesssim M/M_{\odot} \lesssim$ 2.30 and hence should be T-Tauri stars. About half of the weak line T-Tauri stars are found to be fast rotators with a period of $\lesssim$ 2 days as compared to the classical T-Tauri stars. Some of the variables belong to the field star population., Comment: 14pages, 16 figures, accepted for publication in MNRAS. arXiv admin note: text overlap with arXiv:1209.1472

Let $F$ and $G$ be two bounded operators on two Hilbert spaces. Let their numerical radii be no greater than one. This note investigate when there is a $\Gamma$-contraction $(S,P)$ such that $F$ is the fundamental operator of $(S,P)$ and $G$ is the fundamental operator of $(S^*,P^*)$. Theorem 1 puts a necessary condition on $F$ and $G$ for them to be the fundamental operators of $(S,P)$ and $(S^*,P^*)$ respectively. Theorem 2 shows that this necessary condition is sufficient too provided we restrict our attention to a certain special case. The general case is investigated in Theorem 3. Some of the results obtained for $\Gamma$-contractions are then applied to tetrablock contractions to figure out when two pairs $(F_1, F_2)$ and $(G_1, G_2)$ acting on two Hilbert spaces can be fundamental operators of a tetrablock contraction $(A, B, P)$ and its adjoint $(A^*, B^*, P^*)$ respectively. This is the content of Theorem 4.

In this paper we present time series photometry of 104 variable stars in the cluster region NGC 1893. The association of the present variable candidates to the cluster NGC 1893 has been determined by using $(U-B)/(B-V)$ and $(J-H)/(H-K)$ two colour diagrams, and $V/(V-I)$ colour magnitude diagram. Forty five stars are found to be main-sequence variables and these could be B-type variable stars associated with the cluster. We classified these objects as $\beta$ Cep, slowly pulsating B stars and new class variables as discussed by Mowlavi et al. (2013). These variable candidates show $\sim$0.005 to $\sim$0.02 mag brightness variations with periods of $<$ 1.0 d. Seventeen new class variables are located in the $H-R$ diagram between the slowly pulsating B stars and $\delta$ Scuti variables. Pulsation could be one of the causes for periodic brightness variations in these stars. The X-ray emission of present main-sequence variables associated with the cluster lies in the saturated region of X-ray luminosity versus period diagram and follows the general trend by Pizzolato et al. (2003)., Comment: 12pages, 12 figures and 3 tables, Accepted for publication in MNRAS

Continuing the attempt to understand the properties of the stellar content in the young cluster NGC 1893 we have carried out a comprehensive multi-wavelength study of the region. The present study focuses on the X-ray properties of T-Tauri Stars (TTSs) in the NGC 1893 region. We found a correlation between the X-ray luminosity, $L_X$, and the stellar mass (in the range 0.2$-$2.0 \msun) of TTSs in the NGC 1893 region, similar to those reported in some other young clusters, however the value of the power-law slope obtained in the present study ($\sim$ 0.9) for NGC 1893 is smaller than those ($\sim$1.4 - 3.6) reported in the case of TMC, ONC, IC 348 and Chameleon star forming regions. However, the slope in the case of Class III sources (Weak line TTSs) is found to be comparable to that reported in the case of NGC 6611 ($\sim$ 1.1). It is found that the presence of circumstellar disks has no influence on the X-ray emission. The X-ray luminosity for both CTTSs and WTTSs is found to decrease systematically with age (in the range $\sim $ 0.4 Myr - 5 Myr). The decrease of the X-ray luminosity of TTSs (slope $\sim$ -0.6) in the case of NGC 1893 seems to be faster than observed in the case of other star-forming regions (slope -0.2 to -0.5). There is indication that the sources having relatively large NIR excess have relatively lower $L_X$ values. TTSs in NGC 1893 do not follow the well established X-ray activity - rotation relation as in the case of main-sequence stars., Comment: 10 pages, 7 figures, Accepted for publication in New Astronomy

We present results of time series photometry to search for variable stars in the field of metal-poor globular cluster NGC 4590 (M68). Periods have been revised for 40 known variables and no significant changes were found. A considerable change in Blazhko effect for V25 has been detected. Among nine newly discovered variable candidates, 5 stars are of RRc Bailey type variables while 4 stars are unclassified. The variable stars V10, V21, V50 and V51 are found to be cluster members based on the radial velocity data taken from literature., Comment: Accepted for publication to New Astronomy 5 figures, 5 tables

$UBVRI$ photometry of the five open clusters Czernik 4, Berkeley 7, NGC 2236, NGC 7226 and King 12 has been carried out using ARIES 104 cm telescope, Nainital. Fundamental cluster parameters such as foreground reddening $E(B-V)$, distance, and age have been derived by means of the observed two colour and colour-magnitude diagrams, coupled to comparisons with theoretical models. $E(B-V)$ values range from 0.55 to 0.74 mag, while ages derived for these clusters range from $\sim$10 to $\sim$500 Myr. We have also studied the spatial structure, mass function and mass segregation effects. The present study shows that evaporation of low mass stars from the halo of the clusters increases as they evolve., Comment: 19 pages, 12 figures and 7 tables, Accepted for publication in New Astronomy

We present results of multi-epoch (fourteen nights during 2007-2010) $V$-band photometry of the cluster NGC 1893 region to identify photometric variable stars in the cluster. The study identified a total of 53 stars showing photometric variability. The members associated with the region are identified on the basis of spectral energy distribution, $J-H/H-K$ two colour diagram and $V/V-I$ colour-magnitude diagram. The ages and masses of the majority of pre-main-sequence sources are found to be $\lesssim$ 5 Myr and in the range 0.5 $\lesssim$ $M/M_{\odot}$ $\lesssim$ 4, respectively. These pre-main-sequence sources hence could be T Tauri stars. We also determined the physical parameters like disk mass and accretion rate from the spectral energy distribution of these T Tauri stars. The periods of majority of the T Tauri stars range from 0.1 to 20 day. The brightness of Classical T Tauri stars is found to vary with larger amplitude in comparison to Weak line T Tauri stars. It is found that the amplitude decreases with increase in mass, which could be due to the dispersal of disks of massive stars., Comment: 15pages, 16 figures and 5 tables, Accepted for publication in MNRAS

We present time-series photometry of stars located in the extremely young open cluster Berkeley 59. Using the 1.04 m telescope at ARIES, Nainital, we have identified 42 variables in a field of 13'x13' around the cluster. The probable members of the cluster are identified using (V, V-I) colour-magnitude diagram and (J-H, H-K) colour-colour diagram. Thirty one variables are found to be pre-main sequence stars associated with the cluster. The ages and masses of pre-main sequence stars are derived from colour-magnitude diagram by fitting theoretical models to the observed data points. The ages of the majority of the probable pre-main sequence variable candidates range from 1 to 5 Myrs. The masses of these pre-main sequence variable stars are found to be in the range of ~0.3 to ~3.5 Msun and these could be T Tauri stars. The present statistics reveal that about 90% T Tauri stars have periods < 15 days. The classical T Tauri stars are found to have larger amplitude in comparison to the weak line T Tauri stars. There is an indication that the amplitude decreases with increase of the mass, which could be due to the dispersal of disk of relatively massive stars., Comment: 11 pages, 12 Figures and 2 tables, Accepted for publication in MNRAS

We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz basic sequences., Comment: 15 pages

This paper proves two theorems. The first of these simplifies and lends clarity to the previous characterizations of the invariant subspaces of $S$, the operator of multiplication by the coordinate function $z$, on $L^2(\mathbb{T};\mathbb{C}^n)$, where $\mathbb{T}$ is the unit circle, by characterizing the invariant subspaces of $S^n$ on scalar valued $L^p$ ($0

ARIES Baker-Nunn Schmidt telescope project is converting a Baker-Nunn satellite tracking camera for Astronomical research. Original Baker-Nunn camera produces an extremely large (5X30 degree) curved focal plane at the prime focus for photographic imaging. We present here the re-designing of the camera produces a wide (4 X 4 degree) flat field of view for CCD imaging observations, which have many scientific potentials in Astronomy. Imaging performance of the CCD camera is also estimated., Comment: 7 pages, 3 figure, Proceedings of International Conference on Trends in Optics and Photonics(IConTOP), 2009, edited by A. Ghosh and D. Choudhury, held at Kolkata, India during 1-4 March 2009, pp380-386,ISBN 978-81-908188-0-3

We give a short direct proof of Agler's factorization theorem that uses the abstract characterization of operator algebras. the key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional information about these factorizations in the case of polynomials.