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Let a(n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of \(SL_2({\mathbb R})\). This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of \(a(n)a(n+r)\) for a fixed positive integer r.

In this note, we give a counterexample for Theorem 2.3 of the above mentioned paper that is a generalization of the Grothendieck non-vanishing theorem to a class of modules larger than finitely generated modules.

In this short paper, we study the existence of common universal series for uncountable families of specific linear operators. In particular we deal with some derived forms of Seleznev’s theorem and we obtain common universal elements in the space of formal power series in several complex variables.

Let G be a finite nonabelian group, P ∈Sylp(G), and bcl(G) the largest length of conjugacy classes of G. In this short paper, we prove that in general and |P/Op(G)| < bcl(G) in the case where P is abelian.

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrodinger equation $$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$ in $$H^1$$ . The well-posedness theory in $$H^1$$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $$\beta =(4-2\alpha )/(n-2)$$ . It is still an open problem. The main contribution of this paper is to develop the theory in this case.

In this paper, we study the following nonlinear elliptic systems: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1+V_1(x)u_1=\partial _{u_1}F(x,u)&{}\quad x\in {\mathbb {R}}^N,\\ -\Delta u_2+V_2(x)u_2=\partial _{u_2}F(x,u)&{}\quad x\in {\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$ - Δ u 1 + V 1 ( x ) u 1 = ∂ u 1 F ( x , u ) x ∈ R N , - Δ u 2 + V 2 ( x ) u 2 = ∂ u 2 F ( x , u ) x ∈ R N , where $$u=(u_1,u_2):{\mathbb {R}}^N\rightarrow {\mathbb {R}}^2$$ u = ( u 1 , u 2 ) : R N → R 2 , F and $$V_i$$ V i are periodic in $$x_1,\ldots ,x_N$$ x 1 , … , x N and $$0\notin \sigma (-\,\Delta +V_i)$$ 0 ∉ σ ( - Δ + V i ) for $$i=1,2$$ i = 1 , 2 , where $$\sigma (-\,\Delta +V_i)$$ σ ( - Δ + V i ) stands for the spectrum of the Schrödinger operator $$-\,\Delta +V_i$$ - Δ + V i . Under some suitable assumptions on F and $$V_i$$ V i , we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the result in Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009).

During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for $${\mathfrak {L}}_{1,\infty } (H)$$ . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified.

Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.

Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a $$C^\star $$ -algebra onto a Banach algebra. We then use this result to show that a $$C^\star $$ -algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function $$\phi :A\rightarrow B$$ satisfying (i) $$\sigma \left( \phi (x)\phi (y)\phi (z)\right) =\sigma \left( xyz\right) $$ for all $$x,y,z\in A$$ (where $$\sigma $$ denotes the spectrum), and (ii) $$\phi $$ is continuous at $$\mathbf 1$$ . In particular, if (in addition to (i) and (ii)) $$\phi (\mathbf 1)=\mathbf 1$$ , then $$\phi $$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Bresar and Spenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).

Let k be a field of characteristic $$ p\not =0 $$. For a (purely) inseparable extension K / k the notion of modularity, defined by M.E. Sweedler in the 60s, is a very important property, very much like being Galois for a separable extension. We have defined, together with M. Chellali, a generalization of the notion of modularity, called lower quasi-modularity: K / k is lower quasi-modular (lq-modular) if for some finite extension $$k'$$ over k we have that $$K/k'$$ is modular. In subsequent papers M. Chellali and the author have studied various properties of lq-modular field extensions, including the existence of lq-modular closures in case $$[k{:}k^p]$$ is finite. In this paper we prove a similar result, without the hypothesis on k but with extra assumptions on K / k: the extension needs to be q-finite, that is, there must exist an integer M such that for every positive integer n the field $$K\cap k^{p^{-n}}$$ is generated by at most M elements on k. A number of properties of lq-modular closures are determined and examples are presented illustrating the results.

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kahler metric associated with the Kahler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.

In 1933, Lehmer exhibited the polynomial $$\begin{aligned} L(z)=z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1 \end{aligned}$$ with Mahler measure $$\mu _0>1$$ . Then he asked if $$\mu _0$$ is the smallest Mahler measure, not 1. This question became known as the Lehmer conjecture and it was apparently solved in the positive, while this paper was in preparation [19]. In this paper we consider those polynomials of the form $$\chi _A$$ , that is, Coxeter polynomials of a finite dimensional algebra A (for instance $$L(z)=\chi _{\mathbb {E}_{10}}$$ ). A polynomial in $$\mathbb {Z}[T]$$ which is either cyclotomic or with Mahler measure $$\ge \mu _0$$ is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that A being a tree algebra is a sufficient condition for $$\chi _A$$ to be Lehmer.

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**

In this paper, we show that the number of monic integer polynomials of degree \(d \ge 1\) and height at most H which have no real roots is between \(c_1H^{d-1/2}\) and \(c_2 H^{d-1/2}\), where the constants \(c_2>c_1>0\) depend only on d. (Of course, this situation may only occur for d even.) Furthermore, for each integer s satisfying \(0 \le s < d/2\) we show that the number of monic integer polynomials of degree d and height at most H which have precisely 2s non-real roots is asymptotic to \(\lambda (d,s)H^{d}\) as \(H \rightarrow \infty \). The constants \(\lambda (d,s)\) are all positive and come from a recent paper of Bertok, Hajdu, and Pethő. They considered a similar question for general (not necessarily monic) integer polynomials and posed this as an open question.

We revive the concept of Lambda-freeness of Mlotkowski, which describes a mixture of classical and free independence between algebras of random variables. In particular, we give a description of this in terms of cumulants; this will be instrumental in the subsequent paper [SW] where the quantum symmetries underlying these mixtures of classical and free independences will be considered., Comment: We rewrote and shortened the earlier version. The third version contains mainly the results which are new compared to the paper of Mlotkowski

We investigate the possible values for geometric genera of subvarieties in a smooth projective variety. Values which are not attained are called gaps. For curves on a very general surface in $\mathbb{P}^3$, the initial gap interval was found by Xu (see [7] in References), and the next one in our previous paper (see [4] in References), where also the finiteness of the set of gaps was established and an asymptotic upper bound of this set was found. In the present paper we extend some of these results to smooth projective varieties of arbitrary dimension using a different approach., 9 pages, submitted preprint

In a recent paper, we introduced sine functions on commutative hypergroups. These functions are natural generalizations of those functions on groups which are products of additive and multiplicative homomorphisms. In this paper, we describe sine functions on different types of hypergroups, including polynomial hypergroups, Sturm–Liouville hypergroups, etc. A non-commutative hypergroup is also considered.

Let $$I_n(G)$$ denote the number of elements of order n in a finite group G. In 1979, Herzog (Proc Am Math Soc 77:313–314, 1979) conjectured that two finite simple groups containing the same number of involutions have the same order. In a 2018 paper (Arch Math 111:349–351, 2018), Zarrin disproved Herzog’s conjecture with a counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group such that $$I_2(G)=I_2(S)$$ and $$I_p(G) =I_p(S)$$ for some odd prime divisor p, then $$|G|=|S|$$ ”. In this paper, we give more counterexamples to Herzog’s conjecture. Moreover, we disprove Zarrin’s conjecture.

In this paper, we consider the split common null point problem in Banach spaces. Then using the metric resolvents of maximal monotone operators and the metric projections, we prove a strong convergence theorem for finding a solution of the split common null point problem in Banach spaces. The result of this paper seems to be the first one to study it outside Hilbert spaces.

We determine the set of primitive integral solutions to the generalised Fermat equation x^2 + y^3 = z^15. As expected, the only solutions are the trivial ones with xyz = 0 and the non-trivial pair (x,y,z) = (+-3, -2, 1)., The paper is slightly shorter. Specifically, in the notation of the paper, we are now able to carry out Chabauty on the curve C_{3,0}. This allows us to eliminate a lengthy elliptic curve Chabauty computation

We study a Stackelberg strategy subject to the evolutionary linearized Kirchhoff equation for small vibrations of a stretched elastic string when the ends are variables. We assume that we can act in the dynamic of the system by a hierarchy of controls. According to the formulation given by H. von Stackelberg (see [3]), there are local controls, called followers, and global controls, called leaders. In fact, one considers situations where there are two cost (objective) functions. One possible way is to cut the control into two parts, one being thought of as “the leader” and the other one as “the follower”. This situation is studied in the paper, with one of the cost functions being of the controllability type. Existence and uniqueness is proven. The optimality system is given in the paper.

In this paper we consider the following Kirchhoff type problem: $$(\mathcal{K}) \quad \left(1 + \lambda \int\limits_{\mathbb{R}^3}\big(|\nabla u|^2 + V(y)u^2dy\big)\right)[-\Delta u + V(x)u] = |u|^{p-2}u, \quad {\rm in} \, \mathbb{R}^3,$$ where $${p\in (2, 6)}$$ , λ > 0 is a parameter, and V(x) is a given potential. Some existence and nonexistence results are obtained by using variational methods. Also, the “energy doubling” property of nodal solutions of $${(\mathcal{K})}$$ is discussed in this paper.

If a finite group G is \({\mathcal{N}}\)-constrained, then the nilpotent injectors of G form a single conjugacy class of subgroups. In this paper we shall generalize this result. This paper is part of a larger program investigating a special type of nilpotent injectors in arbitrary finite groups.

For a Banach space E and its bidual space E'', the function k(H) defined on bounded subsets H of E measures how far H is from being σ(E,E')-relatively compact in E. This concept, introduced independently by Granero, and Cascales et al., has been used to study a quantitative version of Krein¿s theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper, a quantitative version of Krein¿s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E, the above function k(H) has been defined in thisi paper by menas of d(h,E) is the natural distance of h to E in the bidual E''. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds k(coH) < (2^(n+1) − 2)k(H) + 1/2^n for all n ∈ N. Consequently, this yields also the following formula k(coH) ≤ (k(H))^(1/2))(3-2(k(H)^(1/2))). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein¿s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet spaces. We also define and discuss two other measures of weak non-compactness lk(H) and k'(H) for a Fréchet space and provide two quantitative versions of Krein¿s theorem for both functions., The research was supported for C. Angosto by the project MTM2008-05396 of the Spanish Ministry of Science and Innovation, for J. Kakol by National Center of Science, Poland, Grant No. N N201 605340, and for M. Lopez-Pellicer by the project MTM2010-12374-E (complementary action) of the Spanish Ministry of Science and Innovation.

In this paper, we study a uniqueness question of entire functions sharing an entire function of smaller order with their difference operators. The results in this paper extend Theorem 1.1 in [19] by Liu and Yang and deal with Question 1 in [19], where the entire functions are of finite order. Moreover, we repair certain statements in [21] by Li et al., which in turn had depended on questionable assertions of Lemma 2.6 in [20]. Examples are provided to show that the results in this paper are best possible.

In this paper, we deal with a uniqueness question of entire functions sharing a nonzero complex number with their difference operators. The results in this paper improve Theorem 1.1 in Liu and Yang (Arch. Math. 92 (2009), 270–278) and deal with Question 1 in Liu and Yang (2009), where the entire functions are of finite orders.

Clifford indices for semistable vector bundles on a smooth projective curve of genus at least four were defined in a previous paper of the authors. The present paper studies bundles which compute these Clifford indices. We show that under certain conditions on the curve all such bundles and their Serre duals are generated.

In this paper we find the structure for the automorphism group of a split metacyclic 2-group G. It can be seen as a continuation of the paper (Curran in Arch. Math. 89 (2007), 10–23) and it makes it complete. We propose a different approach to the problem than in the paper (Curran in Arch. Math. 89 (2007), 10–23). Our intention is to show that apart from some cases of 2-groups AutG has a structure similar to that of a direct product of two groups with no common direct factor [which was considered in Bidwell, Curran, and McCaughan (Arch. Math. 86 (2006), 481–489)].

The paper contains the identification of the $$L_p$$ L p norms of paraproducts, defined on general measure spaces equipped with a dyadic-like structure. The proof exploits the Bellman function method.

In 1947, M.S. Macphail constructed a series in $$\ell _{1}$$ that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of Banach space theory, by showing that in all infinite-dimensional Banach spaces, there exists an unconditionally summable sequence that fails to be absolutely summable. More precisely, the Dvoretzky–Rogers theorem asserts that in every infinite-dimensional Banach space E, there exists an unconditionally convergent series $$\sum x^{\left( j\right) }$$ such that $$\sum \Vert x^{(j)}\Vert ^{2-\varepsilon }=\infty $$ for all $$\varepsilon >0$$ . Their proof is non-constructive and Macphail’s result for $$E=\ell _{1}$$ provides a constructive proof just for $$\varepsilon \ge 1$$ . In this note, we revisit Macphail’s paper and present two alternative constructions that work for all $$\varepsilon >0.$$

Let R be an algebraic algebra over an infinite field and $$*$$ be an involution on R. We show that if the units of R, $${\mathcal {U}}(R)$$ , satisfy a $$*$$ -Laurent polynomial identity, then R satisfies a polynomial identity. Also, let G be a torsion group and F a field. As a generalization of Hartley’s Conjecture, in Broche et al. (Arch Math 111:353–367, 2018), it is shown that if $${\mathcal {U}}(FG)$$ satisfies a Laurent polynomial identity which is not satisfied by the units of the relative free algebra $$F[\alpha , \beta :\alpha ^2=\beta ^2=0]$$ , then FG satisfies a polynomial identity. In this paper, we instead consider non-torsion groups G and provide some necessary conditions for $${\mathcal {U}}(FG)$$ to satisfy a Laurent polynomial identity.

A natural variant of Lehmer’s conjecture that the Ramanujan $$\tau $$ -function never vanishes asks whether, for any given integer $$\alpha $$ , there exist any $$n \in \mathbb {Z}^+$$ such that $$\tau (n) = \alpha $$ . A series of recent papers excludes many integers as possible values of the $$\tau $$ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $$\tau (n)$$ . We synthesize these results and methods to prove that if $$0< \left| \alpha \right| < 100$$ and $$\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$$ , then $$\tau (n) \ne \alpha $$ for all $$n > 1$$ . Moreover, if $$\alpha \in T$$ and $$\tau (n) = \alpha $$ , then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that $$\left| \tau (n) \right| > 100$$ for all $$n > 2$$ .

Foxby (Math Scand 2:175–186, 1971–1972) showed that a Cohen-Macaulay module over a Gorenstein local ring has finite projective dimension if and only if its canonical module has finite injective dimension. In this paper, we establish the result given by Foxby in a general setting. As a byproduct, some criteria to detect the Cohen-Macaulay property of a ring are provided in terms of intrinsic properties of certain local cohomology modules. Also, as an application, we show that any Cohen-Macaulay module that has a canonical module with finite injective dimension satisfies the Auslander–Reiten conjecture.

We deal with integrals of invariant measures of pairs of planes in euclidean space $\mathbb{E}^3$ as considered by Hug and Schneider. In this paper we express some of these integrals in terms of functions of the visual angle of a convex set. As a consequence of our results we evaluate the deficit in a Crofton-type inequality due to Blashcke., 16 pages

In this paper, we obtain a new Cheeger–Gromoll splitting theorem on a complete Riemannian manifold admitting a smooth vector field such that its Bakry–Emery Ricci tensor is non-negative and the vector field tends to zero at infinity. The result generalizes the classical Cheeger–Gromoll splitting theorem and the splitting type results of Lichnerowicz, Wei–Wylie, Fang–Li–Zhang, Wylie, Khuri–Woolgar–Wylie, Lim, and more.

In the previous paper [9] the author proved the joint limit theorem for the Riemann zeta function and the Hurwitz zeta function attached with a transcendental real number. As a corollary, the author obtained the joint functional independence for these two zeta functions. In this paper, we study the joint value distribution for the Riemann zeta function and the Hurwitz zeta function attached with an algebraic irrational number. Especially we establish the weak joint functional independence for these two zeta functions.

In this paper, we study the growth of solutions of a first order linear differential equation. From this we verify that a conjecture given by Bruck is true under the restriction of the hyper order less than 1/2, and obtain some uniqueness theorems of a nonconstant entire function and its first derivative sharing a nonzero constant CM. The results in this paper also improve some known results. Some examples show that the results in this paper are best possible.

The proof of the existence and of the uniqueness of groups of J3-type by G. Higman and J. McKay is based on the fact that a group of J3-type is a faithful completion of an amalgam of J3-type, see [11]. In this paper here, we provide a direct reference for that fact. The proofs in this paper are elementary and we do not use any character theory.

Minimizers of the total variation subject to a prescribed $$ \user1{\mathcal{L}}^1 $$ -norm are considered as eigensolutions of the 1-Laplace operator. The derivation of the corresponding eigenvalue equation, which requires particular care due to the lack of smoothness, is carried out in a previous paper by using particular methods of nonsmooth analysis. The present paper provides a simpler proof that exploits the special structure of the problem.

This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations. Let $$D: = {\sum\limits_{i = 1}^n {\frac{\partial }{{\partial x_{i} }}e_{i}}}$$ be the Euclidean Dirac operator in the n-dimensional flat space $$\mathbb{R}^{n},\;{\mathbf{E}}: = {\sum\limits_{i = 1}^n {x_{i} \frac{\partial }{{\partial x_{i} }}}}$$ the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe explicitly the structure of the solutions to the PDE system $$\left[ {D - \lambda - (1 + \alpha )\frac{{\text{x}}} {{\text{|x|}}^{2}}{\mathbf{E}}} \right]f = 0$$ in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials.

In this paper, we analyse oscillatory properties of a general class of linear difference equations. Applying the modified Riccati technique, we prove an oscillation criterion for the studied equations and we formulate its consequences. In contrast to many known criteria, in the presented results, there are not considered any auxiliary sequences. The results are based directly on the coefficients of the treated equations, i.e., the obtained results are easy to use. In addition, recently, we have proved a non-oscillatory counterpart of the presented criterion. The combination implies that the studied equations are conditionally oscillatory.

In this paper, we study gradient estimates of positive smooth solutions to the p-Laplace equation $$\begin{aligned} (p-1)^{p-1}\Delta _pu+au^{p-1}\log u^{p-1}=0, \end{aligned}$$ which is related to the $$L^p$$ -log-Sobolev constant on Riemannian manifolds, where a is a nonzero constant. As applications, some Liouville type results are provided.

A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper, we give a lower bound for the coclass of finite nonabelian p-groups G having no noninner automorphism of order p leaving the Frattini subgoup $$\Phi (G)$$ elementwise fixed. As a consequence, the verification of the conjecture is reduced to the case of finite nonabelian p-groups G in which the coclass of G is greater than the minimum number of generators of G.

In this paper, we study the problem of the existence of left invariant Ricci flat metrics on nilpotent Lie groups. We mainly prove that any nilpotent Lie algebra obtained by a double extension of an Abelian Lie algebra admits at least one left invariant Ricci flat metric. As an application, we obtain certain new nilpotent Lie algebras which admit left invariant Ricci flat metrics.

Tartar gave an alternative proof of the Riesz–Thorin interpolation theorem for operators of strong types (1, 1) and $$(\infty ,\infty )$$ . His method characterizes the $$L^{p}$$ norm in terms of the Lebesgue spaces $$L^{1}$$ and $$L^{\infty }$$ , and works not only for complex Lebesgue spaces but also for real Lebesgue spaces. The aim of this paper is to extend the proof for operators of strong types $$(p_{1},q_{1})$$ and $$(\infty ,\infty )$$ with $$1\le p_{1}\le q_{1}

Let $${{\mathcal {A}}}$$ be a Banach algebra and let $$\varphi $$ be a non-zero character on $${{\mathcal {A}}}$$ . Suppose that $${{\mathcal {A}}}_M$$ is the closure of the faithful Banach algebra $${{\mathcal {A}}}$$ in the multiplier norm. In this paper, topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}_M^*$$ are defined and studied. Under some conditions on $${{\mathcal {A}}}$$ , we will show that the set of topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}^*$$ and on $${{\mathcal {A}}}_M^*$$ have the same cardinality. The main applications are concerned with the quantum group algebra $$L^1({\mathbb {G}})$$ of a locally compact quantum group $${\mathbb {G}}$$ . In particular, we obtain some characterizations of compactness of $${\mathbb {G}}$$ in terms of the existence of a non-zero (weakly) compact left or right multiplier on $$L^1_M({\mathbb {G}})$$ or on its bidual in some senses.