Keywords: Consciousness, Time, AI, Relativity, Quantum Mechanics, Reality,
Responsible AI

# Unifying Consciousness and Time to Enhance Artificial Intelligence

Mahendra Samarawickrama<EMAIL_ADDRESS>Centre for Consciousness
Studies, Australia

###### Abstract

Consciousness is a sequential process of awareness which can focus on one
piece of information at a time. This process of awareness experiences
causation which underpins the notion of time while it interplays with matter
and energy, forming reality. The study of Consciousness, time and reality is
complex and evolving fast in many fields, including metaphysics and
fundamental physics. Reality composes patterns in human Consciousness in
response to the regularities in nature. These regularities could be physical
(e.g., astronomical, environmental), biological, chemical, mental, social,
etc. The patterns that emerged in Consciousness were correlated to the
environment, life and social behaviours followed by constructed frameworks,
systems and structures. The complex constructs evolved as cultures, customs,
norms and values, which created a diverse society. In the evolution of
responsible AI, it is important to be attuned to the evolved cultural, ethical
and moral values through Consciousness. This requires the advocated design of
self-learning AI aware of time perception and human ethics.

##  I Introduction

The notion of time is an integral part of consciousness [1]. The consciousness
experiences the causation or changes in reality/environment and perceives the
time. Therefore, in our previous publication [2], we assumed that
consciousness is a sequential process which is aware of a single piece of
information at a time. Even though the brain processes sensory data of five
sensors (i.e., Sight, Sound, Smell, Taste, and Touch) in parallel in the
neural network, the awareness of causation is a sequential process following
cause and effect. See the illustration of this idea in Figure 1,

Figure 1: The interplay of five sensors, brain and consciousness. The brain
processes sensory information in parallel. However, the awareness of causation
(i.e., consciousness) is a sequential process focusing on a single piece of
information at a time. This sequential process of awareness in consciousness
operates fast and consistently, which underpins our perception of reality.

The assumption of sequential awareness in consciousness enables mapping the
perception of time into consciousness. Based on the theory of relativity [3],
the perception of time is relative to the frame of reference. Einstein assumed
that the speed of light is constant in all frames of reference, and the time
is derived based on that fundamental assumption. In our paper, we defined the
shortest time to be aware of reality as a consciousness cycle. Then based on
relativity, this consciousness cycle is also subjected to dilation, like
relativistic time

$T_{v}=\frac{T_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\,,$ (1)

where, $T_{v}$ is the dilated period of the consciousness cycle related to the
rest period of the consciousness cycle $T_{0}$. Note that the
$\sqrt{1-\frac{v^{2}}{c^{2}}}$ is the Lorentz factor, where $v$ is the
relative velocity between inertial reference frames, and $c$ is the speed of
light in a vacuum. Then, we mathematically modelled [2] how consciousness
would interplay with matter and energy, forming reality, which can be adapted
to understand limitations and opportunities in AI consciousness. This paper
extends our discussion towards the time perception of artificial intelligence
systems (AIS).

##  II The notion of time in perception and reality

Humans, like any other life forms, experience time through causation. Patterns
are composed in the human consciousness in response to the regularities in
nature [4]. Since the beginning of human civilisation, humans have learnt and
evolved complex concepts and constructs by incorporating time emerged through
patterns in the consciousness. The earth’s rotation around itself determines
the day, and orbiting around the sun determines the year. The Moon takes about
one month to orbit the earth. The tilt of the earth’s spin axis with respect
to its orbital plane causes the weather seasons. These environmental patterns
cause many biological patterns and lifestyle patterns in human life. To
predict and organise these patterns effectively, humans introduce standard
time with clocks, calendars and various other frameworks. These artificial
frameworks enable us to model time and objectively measure subjective
experiences.

Physics has been evolved by observation of nature with various frameworks of
time. In this way, time became an essential construct and dimension of our
understanding of reality. For example, Newtonian physics [5] evolved assuming
that time is absolute and flows consistently from past to present and into the
future. That enables the development of mathematical models for explaining
patterns in reality with time. However, later observations, such as the
perihelion motion of Mercury, allow humans to understand time as a
relativistic measure rather than an absolute. The modern understanding of the
universe is based on the theory of relativity [6, 7], which is completely
articulated by space-time principles. Based on relativity, John Wheeler [8]
stated, “Space tells matter how to move. Matter tells space how to curve”.
Relativity enables us to accurately understand and predict the behaviours of
black holes, stars, and planets. Further, relativity enables humans to develop
technologies like the atomic clock [9] and Global Positioning System (GPS)
[10] that are useful in everyday life.

The behaviour of particles is completely different to larger objects like
planets, stars, etc. This led to the evolution of Quantum physics [11] as
opposed to relativity. Quantum physics exhibits amazing accuracy in predicted
results in particle physics. However, it greatly disturbs the notion of time
modelled in relativity. For example, in the collapse of the wave function in
quantum entanglement, Einstein described that as a spooky action at a distance
[12]. As per relativity, information cannot transfer faster than the speed of
light. As per the recent discoveries in quantum entanglement, information can
be transferred instantly, faster than the speed of light, making our reality
non-local [13]. The non-local reality contradicts relativity, which is now
applied in quantum teleportation at the subatomic level. On the other hand, at
the quantum level, the reality is uncertain, as described by Heisenberg’s
uncertainty principle [14]. As per the uncertainty principle, it is impossible
to precisely measure or be aware of the position and speed of a particle in a
given time. This brings the limitation of human awareness and perception of
time. Therefore, many believe now that consciousness is fundamental and that
time and causation are derived from consciousness [15].

##  III The implication of principles of time for AIS

The inability to consolidate quantum physics and the theory of relativity
makes our understanding of reality incomplete. Moreover, the new discoveries
proving the idea of non-local reality shake the status quo of fundamental
physics [16]. Therefore, it is still impossible to supervise AI to experience
the notion of time to understand reality precisely. On the other hand, human
understanding of reality is also about 5%, whereas most of the universe
consists of dark matter and dark energy, which humans do not understand [17].
Under these conditions, AI might be used to explore reality and time in a way
we have never imagined. Perhaps incorporating AI to understand reality and
causation might help humans to become fully aware of reality by overcoming
inherent biases from evolution, culture and nature.

Typical Reinforcement Learning (RL) technique can be adapted to automate the
learning of AI. The RL process can be mathematically formulated using Markov
Decision Process (MDP) [18]. That is a sequential learning process by trial
and error. In this process, the learning agent (i.e., AI) sequentially
interacts with the environment with an intelligent decision (i.e. action)
followed by receiving a reward or a penalty based on the policy imposed. There
will be no influence on the AI agent’s action, but convey the value of its
action through feedback with reward or penalty. This way, the AI agent will
self-learn about the environment over time. The RL process is illustrated in
Figure 2:

Figure 2: Components of the Markov Decision Process (MDP) and its function in
the agent-environment interaction. The sequential step of time is represented
by $t$.

##  IV The implication of human beliefs, values and cultures for the
perception of time in AIS

Human beliefs, customs, culture and values are tightly linked with various
dynamics and interpretations of the time and periodicities based on the
movement of the earth, Moon and other terrestrial bodies. From the beginning,
humans identified that time affects life and nature differently. Therefore, in
the Greece era, early Western culture, there were at least three gods
representing different time forms: Chronos, Aion, and Kairos [19]. Chronos
represented the linear time flowing from past to present into the future. This
is the time that humans feel when life passes. In contrast, Aion represented
the cyclical nature of time experienced from natural events such as weather
patterns, rebirths, etc. The third god Kairos represented the opportunist
time, which reflects the appropriate time to achieve a task. In this way,
time, environment and beliefs were tightly linked with life and governed
society and values.

On the other hand, in Eastern culture, the horoscope is one good example of a
planetary and constellation framework underpinning Astrology as a foundation
of certain belief systems [20]. These beliefs assume that Astrology is
associated with time and causality, which can predict the future and guide
humans.

The human observation of the night sky led to perceiving time from various
cyclical patterns going far back in time. For example, the Aboriginal
Australians [21] observed the night sky and mapped them to the environment and
life stages that evolved various customs, arts and even religions. Not only by
interacting planets and stars but the tilt of the earth’s spin axis also
significantly led to diversifying human cultures based on seasons,
particularly when moving away from the equator.

The notion of time and associated beliefs, customs, and values are important
to consider when training AIS [22]. That will help promote human cultural
values, ethics, and diversity, equity and inclusion (DEI). AI development may
need to pay attention to and integrate the time attributes that emerged from
nature, values and cultures. Humans may include them in the policies for
rewarding self-learning AI algorithms (e.g., in MDP).

##  V The implication of biological time on AIS

The biological cycles play a fundamental role in human behaviours and the
perception of time—for example, mood cycles, circadian rhythms, and the
menstrual cycle. Without understanding these biological time-keeping
processes, AI cannot seamlessly integrate with human society when creating
values in health, culture, art, etc. These insights are essential to realising
emotional intelligence, empathy and awareness in AI. Literature shows the
effective use of Cyclic Hidden Markov Models (CyH-MMs) for detecting and
modelling cycles in a multidimensional heterogeneous biological time series
data collection [23]. It is important to attribute the relevant features of
biological processes when training AIS, which raises more awareness about
humans.

Recent discoveries in quantum physics argue that our reality is non-local,
where awareness can happen instantly, faster than the speed of light.
Physicists and neurologists think brain neurons might be aware of the quantum
world through the orchestrated collapse of microtubules in the neurons in the
brain [24, 25]. If this hypothesis is true, then there are possibilities that
human awareness can be linked with non-local realities to expand our
consciousness across the universe instantly. From this perspective, future AI
might need to be evolved with the capabilities of biological neurons, which
interplay with the quantum realities. The recent development of neurotech
realising brain-computer interface (BCI) along with emerging quantum computers
might enable such capabilities in the near future [26].

##  VI Conclusion

Consciousness and perception of time and causation are key to awareness and
understanding reality. The notion of time emerged from causation, a perception
relative to the observer as per the relativity principles. In relativity, it’s
not time but the light-speed constant in all frames of reference. In contrast,
in quantum entanglement, the reality is non-local, and information can be
transferred instantly faster than light. While the principles of time
contradict the foundation of physics, time also influenced the formation of
diverse customs, values and cultures based on patterns that emerged from
nature, particularly around the regularities in the earth’s movement,
environment, astronomy and biology. Therefore, understanding time and related
artefacts (i.e., cultures, beliefs, values, customs, physics, health, etc.)
are very important to realise deep awareness of reality. From the AIS
perspective, it will enhance the understanding of AI in human health,
cultures, customs, values and various other diversities. Bringing this
awareness to AI will be a challenging and complex yet rewarding milestone in
the evolution of ethical and responsible AI.

## References

  * [1] L. Kent and M. Wittmann, “Erratum to: Time consciousness: the missing link in theories of consciousness,” Neuroscience of Consciousness, vol. 2021, 05 2021.
  * [2] M. Samarawickrama, “Unifying Matter, Energy and Consciousness,” International Conference on Mathematical Modeling in the Physical Sciences (IC-MSQUARE), 9 2022.  https://youtu.be/Pby1TfEluqE.
  * [3] A. Einstein, “Zur Elektrodynamik bewegter Körper. (German) [On the electrodynamics of moving bodies],” Annalen der Physik, vol. 322, no. 10, pp. 891–921, 1905.
  * [4] S. Blackburn, “Hume and thick connexions,” Philosophy and Phenomenological Research, vol. 50, pp. 237–250, 1990.
  * [5] I. Newton, Philosophiae Naturalis Principia Mathematica.  London: Royal Society, 1687.
  * [6] A. Einstein, “Zur allgemeinen Relativitätstheorie. (German) [Toward a General Theory of Relativity],” j-S-B-PREUSS-AKAD-WISS-2, pp. 778–786, 799–801, 1915.
  * [7] A. Einstein, “Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie. (German) [Explanation of the perihelical motion of Mercury from the General Theory of Relativity],” j-S-B-PREUSS-AKAD-WISS-2, pp. 831–839, 1915.
  * [8] J. A. Wheeler, “Geometrodynamics and the Problem of Motion,” Reviews of Modern Physics, vol. 33, pp. 63–78, Jan 1961.
  * [9] N. F. Ramsey, “History of early atomic clocks,” Metrologia, vol. 42, p. S1, jun 2005.
  * [10] R. Maddison and C. N. Mhurchu, “Global positioning system: a new opportunity in physical activity measurement,” International Journal of Behavioral Nutrition and Physical Activity, vol. 6, no. 1, p. 73, 2009.
  * [11] P. A. Zyla et al. (Particle Data Group), “Review of Particle Physics,” Progress of Theoretical and Experimental Physics, vol. 2020, 08 2020.  083C01.
  * [12] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev., vol. 47, pp. 777–780, May 1935.
  * [13] A. Lohrey and B. Boreham, “The nonlocal universe,” Communicative & Integrative Biology, vol. 13, no. 1, pp. 147–159, 2020.
  * [14] W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. (German) [On the ideological content of quantum theoretical kinematics and mechanics],” Z. Physik, vol. 43, pp. 172–198, Mar. 1927.
  * [15] D. D. Hoffman, “The Origin of Time In Conscious Agents,” Cosmology, vol. 18, pp. 494–520, 2014.  https://www.cogsci.uci.edu/ ddhoff/HoffmanTime.pdf.
  * [16] W. Sulis, “Locality Is Dead! Long Live Locality!,” Frontiers in Physics, vol. 8, 2020.
  * [17] E. Oks, “Brief review of recent advances in understanding dark matter and dark energy,” New Astronomy Reviews, vol. 93, p. 101632, 2021.
  * [18] M. van Otterlo and M. Wiering, “Reinforcement Learning and Markov Decision Processes,” Reinforcement Learning: State-of-the-Art, pp. 3–42, 2012\.
  * [19] J. E. Smith, “Time, Times, and the ‘Right Time’; Chronos and Kairos,” The Monist, vol. 53, no. 1, pp. 1–13, 1969.
  * [20] N. Campion, “Astrology as cultural astronomy,” Handbook of Archaeoastronomy and Ethnoastronomy, pp. 103–116, 2015.
  * [21] D. W. Hamacher, “Comet and meteorite traditions of aboriginal australians,” Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, pp. 1–4, 2008.
  * [22] K. Lee and K. Joshi, “Understanding the Role of Cultural Context and User Interaction in Artificial Intelligence Based Systems,” Journal of Global Information Technology Management, vol. 23, no. 3, pp. 171–175, 2020\.
  * [23] E. Pierson, T. Althoff, and J. Leskovec, “Modeling Individual Cyclic Variation in Human Behavior,” in Proceedings of the 2018 World Wide Web Conference, p. 107–116, 2018.
  * [24] S. Hameroff and R. Penrose, “Consciousness in the universe: A review of the ‘Orch OR’ theory,” Physics of Life Reviews, vol. 11, no. 1, pp. 39–78, 2014.
  * [25] S. Hameroff, “‘Orch OR’ is the most complete, and most easily falsifiable theory of consciousness,” Cognitive Neuroscience, vol. 12, no. 2, pp. 74–76, 2021.
  * [26] S. Saha, K. A. Mamun, K. Ahmed, R. Mostafa, G. R. Naik, S. Darvishi, A. H. Khandoker, and M. Baumert, “Progress in Brain Computer Interface: Challenges and Opportunities,” Frontiers in Systems Neuroscience, vol. 15, Feb 2021.


# Weighted frames, weighted lower semi frames and unconditionally convergent
multipliers

P. Balazsa,1, R. Corsob,2, D. Stoevac,3

(aAcoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse
12-14, 1040 Vienna, Austria
bDipartimento di Matematica e Informatica, Università degli Studi di Palermo,
Via Archirafi 34,
90123 Palermo, Italy
cFaculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090
Vienna, Austria<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
October 21, 2023 )

###### Abstract

In this paper we ask when it is possible to transform a given sequence into a
frame or a lower semi frame by multiplying the elements by numbers. In other
words, we ask when a given sequence is a weighted frame or a weighted lower
semi frame and for each case we formulate a conjecture. We determine several
conditions under which these conjectures are true. Finally, we prove an
equivalence between two older conjectures, the first one being that any
unconditionally convergent multiplier can be written as a multiplier of Bessel
sequences by shifting of weights, and the second one that every
unconditionally convergent multiplier which is invertible can be written as a
multiplier of frames by shifting of weights. We also show that these
conjectures are also related to one of the newly posed conjectures.

Keywords: weighted frames, weighted lower semi frames, multipliers,
unconditionally convergent multipliers.

##  1 Introduction

Let ${\cal H}$ be an infinite dimensional separable Hilbert space with inner
product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$. Throughout the paper
we use $\mathbb{N}$ as an index set, also implicitly. Let $(f_{n})$ be a
sequence with elements of ${\cal H}$. Recall that $(f_{n})$ is called a frame
for ${\cal H}$ if there exist positive constants $A$ and $B$ such that

$A\|f\|^{2}\leq\sum_{n}|\langle f,f_{n}\rangle|^{2}\leq
B\|f\|^{2},\qquad\forall f\in{\cal H}.$ (1)

Frames have been introduced by Duffin and Schaffer [24] and they provide
reconstruction formulas on the entire Hilbert space. More precisely, if
$(f_{n})$ is a frame for ${\cal H}$, then there exists a frame $(g_{n})$ for
${\cal H}$ such that

$f=\sum_{n}\langle f,f_{n}\rangle g_{n}=\sum_{n}\langle f,g_{n}\rangle
f_{n},\qquad\forall f\in{\cal H},$

see [24]. Separate consideration of the inequalities in (1) has also been of
interest. The sequence $(f_{n})$ is a Bessel sequence in ${\cal H}$ with
Bessel bound $B\in(0,\infty)$ if the upper inequality in (1) holds for every
$f\in{\cal H}$. For more on frames and Bessel sequences, we refer e.g. to the
books [17, 26]. The sequence $(f_{n})$ is called a lower semi frame for ${\cal
H}$ with bound $A>0$ if the lower inequality in (1) holds for every $f\in{\cal
H}$. A lower semi frame $\mathcal{F}=(f_{n})$ for ${\cal H}$ provides a
reconstruction formula of the type

$f=\sum_{n}\langle f,f_{n}\rangle g_{n},\qquad\forall f\in{\cal
D}(C_{\mathcal{F}}),$ (2)

where $(g_{n})$ is some Bessel sequence in ${\cal H}$ and ${\cal
D}(C_{\mathcal{F}})$ is the domain of the analysis operator111See Section 2
for the definition of the analysis operator. of $(f_{n})$. This result dates
back to [15]. Contrary to the frame case, where reconstruction of all the
elements of the Hilbert space is guaranteed, for certain lower semi frames a
reconstruction formula may hold only on a strict subset of ${\cal H}$, see
[15, 33]. For more on lower frame sequences, we refer to [1, 2, 15, 18, 22,
33, 34]. For the purpose of a notion symmetrical to the lower semi frame one,
an upper semi-frame for ${\cal H}$ has been introduced [1] to be a Bessel
sequence in ${\cal H}$ that is furthermore complete in ${\cal H}$.

This paper is centered on the following notions.

######  Definition 1.1.

A sequence $(f_{n})$ in ${\cal H}$ is called a weighted frame222Weighted
frames were previously introduced and studied in [29, 14, 6]. The notion is
also connected to scalable frames [31]. (resp. weighted lower semi frame) for
${\cal H}$ if there exists $(\omega_{n})\subset(0,+\infty)$ such that
$(\omega_{n}f_{n})$ is a frame (resp. lower semi frame) for ${\cal H}$.

Specifically, in this paper we study the following problems

  1. P1)

Under which conditions is a given sequence a weighted lower semi frame for
${\cal H}$?

  2. P2)

Under which conditions is a given sequence a weighted frame for ${\cal H}$?

Further introductory definitions and results are given in Sect. 2. Weighted
lower semi frames and Problem P1 are studied in Sect. 3. Similarly, we
dedicate Sect. 4 to weighted frames and to Problem P2. At the end of Sect. 4
we relate the topic to unconditionally convergent multipliers. In particular,
we establish the equivalence between two conjectures put forward in [38]. We
show the surprising fact that if it is true that any unconditionally
convergent invertible multiplier is a multiplier of frames by shifts of
weights, then it is also true that any unconditionally convergent multiplier
is a multiplier of Bessel sequences by shifts of weights.

##  2 Preliminaries

Given an operator $T$ on a vector space we write $\mathcal{D}(T)$ for its
domain and $R(T)$ for its range. As usual, the symbol $\mathcal{B}({\cal H})$
stands for the set of all bounded operators with domain ${\cal H}$ and
$GL({\cal H})$ denotes the set of all bounded bijective operators from ${\cal
H}$ onto ${\cal H}$. A sequence $(f_{n})$ of elements of ${\cal H}$ is
complete in ${\cal H}$ if its linear span is dense in ${\cal H}$. A sequence
$(f_{n})$ in ${\cal H}$ is called minimal if $f_{j}\notin$
$\overline{\text{span}}(f_{n})_{n\neq j}$ for all $j$. As it is well known,
$(f_{n})$ is minimal if and only if it is biorthogonal to some sequence
$(g_{n})$, i.e. $\langle f_{n},g_{m}\rangle=\delta_{n,m}$; a minimal sequence
that is complete in ${\cal H}$ has a unique biorthogonal in ${\cal H}$.
Throughout the paper, we will make use of two operators associated to a
sequence $\mathcal{F}:=(f_{n})$. The analysis operator $C_{\mathcal{F}}:{\cal
D}(C_{\mathcal{F}})\subseteq{\cal H}\to\ell^{2}$ is given by ${\cal
D}(C_{\mathcal{F}})=\\{f\in{\cal H}:\sum_{n}|\langle
f,f_{n}\rangle|^{2}<\infty\\}$ and $C_{\mathcal{F}}f=(\langle
f,f_{n}\rangle)$, for $f\in{\cal D}(C_{\mathcal{F}})$. The synthesis operator
$D_{\mathcal{F}}:{\cal D}(D_{\mathcal{F}})\subseteq\ell^{2}\to{\cal H}$ is
given by ${\cal
D}(D_{\mathcal{F}})=\left\\{(c_{n})\in\ell^{2}:\sum_{n}c_{n}f_{n}\text{
converges in }{\cal H}\right\\}$ and
$D_{\mathcal{F}}(c_{n})=\sum_{n}c_{n}f_{n}$, for $(c_{n})\in{\cal
D}(D_{\mathcal{F}})$. The operator $C_{\mathcal{F}}$ is always closed [15] and
$C_{\mathcal{F}}=D_{\mathcal{F}}^{*}$ [11]. Detailed properties of the
analysis and synthesis operators (alongside other operators associated to
sequences) can be found in [11].

Let us recall some basic results from frame theory (that can be found e.g. in
[17]). For a Bessel sequence $(f_{n})$ and $(c_{n})\in\ell^{2}$ the series
$\sum_{n=1}^{\infty}c_{n}f_{n}$ is always convergent. A Riesz basis $(f_{n})$
for ${\cal H}$ is a complete sequence in ${\cal H}$ satisfying

$A\sum_{n}|c_{n}|^{2}\leq\left\|\sum_{n}c_{n}f_{n}\right\|^{2}\leq
B\sum_{n}|c_{n}|^{2},\qquad\forall(c_{n})\in\ell^{2},$

for some positive constants $A$ and $B$. A sequence is a Riesz basis for
${\cal H}$ if it is a frame for ${\cal H}$ that has a biorthogonal sequence.
As done for frames and lower semi frames, we say that a sequence $(f_{n})$ is
a weighted Riesz basis for ${\cal H}$ if there exists
$(\omega_{n})\subset(0,+\infty)$ such that $(\omega_{n}f_{n})$ is a Riesz
basis for ${\cal H}$. Note that Problem P2 with respect to weighted Riesz
bases is related to the following well-known characterization of Riesz bases.

######  Theorem 2.1.

[17, Lemma 3.6.9] A sequence $(f_{n})$ is a Riesz basis for ${\cal H}$ if and
only if it is an unconditional basis for ${\cal H}$ and it is norm-semi-
normalized (i.e. $0<\inf\|f_{n}\|\leq\sup\|f_{n}\|<\infty$). In other words, a
sequence $(f_{n})$ is a weighted Riesz basis for ${\cal H}$ if and only if it
is an unconditional basis for ${\cal H}$.

In [32, 9] reproducing pairs have been introduced and studied as a way to go
beyond the frame reconstruction formulas. A pair of sequences
$((f_{n}),(g_{n}))$ is called a reproducing pair for ${\cal H}$ if the
operator $T$ defined weakly by $\langle Tf,g\rangle=\sum_{n}\langle
f,f_{n}\rangle\langle g_{n},g\rangle$ belongs to $GL({\cal H})$. Reproducing
pairs are related to the more general notion of sesquilinear forms. For study
on sesquilinear forms, we refer to [30, 18, 22].

##  3 Weighted lower semi frames

Since a weighted frame is, in particular, a weighted lower semi frame, we
start by analyzing sequences that can be transformed into lower semi frames.
First, observe that if a sequence is a weighted lower semi frame for ${\cal
H}$, then it is complete in ${\cal H}$. We conjecture that completeness is
furthermore a sufficient condition for the lower semi frame property:

######  Conjecture 3.1.

Let $(f_{n})$ be a complete sequence in ${\cal H}$ and $A\in(0,\infty)$. There
exists $(\omega_{n})\subset(0,\infty)$ such that $(\omega_{n}f_{n})$ is a
lower semi frame for ${\cal H}$ with a bound $A$.

In this section we determine some classes of sequences for which Conjecture
3.1 holds. First of all, we prove that the corresponding problem for Bessel
sequences can be easily solved and it is helpful for dealing with special
cases of the above conjecture.

######  Lemma 3.2.

Let $(f_{n})$ be a sequence in ${\cal H}$. For any $B>0$ there exists a
sequence $(\lambda_{n})\subset(0,\infty)$ such that $(\lambda_{n}f_{n})$ is a
Bessel sequence with bound $B$.

###### Proof.

Let $B>0$ and let $(\tau_{n})\subset(0,\infty)$ be a sequence such that
$\sum_{n}\tau_{n}^{2}=1$. Define $\lambda_{n}=\tau_{n}\sqrt{B}\|f_{n}\|^{-1}$
if $f_{n}\neq 0$ and $\lambda_{n}=1$ (or any other positive number) if
$f_{n}=0$. Then, by the Cauchy-Schwarz inequality, we have for all $f\in{\cal
H}$ that

$\sum_{n}|\langle
f,\lambda_{n}f_{n}\rangle|^{2}\leq\sum_{n}\tau_{n}^{2}B\|f\|^{2}=B\|f\|^{2}.\qed$

With this lemma we can then prove our first result.

######  Proposition 3.3.

Let $((f_{n}),(g_{n}))$ be a reproducing pair for ${\cal H}$. Then there are
weights $(\lambda_{n})\subset(0,\infty)$ and $(\beta_{n})\subset(0,\infty)$
such that $(\frac{1}{\lambda_{n}}f_{n})$ and $(\frac{1}{\beta_{n}}g_{n})$ are
lower semi-frames for ${\cal H}$, and $(\lambda_{n}g_{n})$ and
$(\beta_{n}f_{n})$ are upper semi-frames for ${\cal H}$. In particular,
Conjecture 3.1 is true for $(f_{n})$ and $(g_{n})$.

###### Proof.

Let $T\in GL({\cal H})$ be the operator such that $\langle
Tf,g\rangle=\sum_{n}\langle f,f_{n}\rangle\langle g_{n},g\rangle$ for all
$f,g\in{\cal H}$. Then

$\langle f,g\rangle=\sum_{n}\langle f,f_{n}\rangle\langle
h_{n},g\rangle,\qquad\forall f,g\in{\cal H},$ (3)

where $(h_{n})=(T^{-1}g_{n})$.

Let $A>0$. By Lemma 3.2, there exists a sequence
$(\lambda_{n})\subset(0,+\infty)$ such that $(\lambda_{n}h_{n})$ is Bessel
with bound $A^{-1}$. Since

$\langle f,g\rangle=\sum_{n}\langle
f,\lambda_{n}^{-1}f_{n}\rangle\langle\lambda_{n}h_{n},g\rangle,\qquad\forall
f,g\in{\cal H},$

using the Cauchy-Schwarz inequality, similar to [17, Lemma 6.3.2] we get that
for every $f\in{\cal H}$,

$\|f\|^{4}\leq(\sum_{n}|\langle
f,\lambda_{n}^{-1}f_{n}\rangle|^{2})(\sum_{n}|\langle
f,\lambda_{n}h_{n}\rangle|^{2})\leq A^{-1}\|f\|^{2}\sum_{n}|\langle
f,\lambda_{n}^{-1}f_{n}\rangle|^{2}.$

Thus, $(\lambda_{n}^{-1}f_{n})$ is a lower semi frame for ${\cal H}$ with
bound $A$. Since $(\lambda_{n}h_{n})$ is a Bessel sequence, clearly
$(\lambda_{n}g_{n})$ is also Bessel.

Further, by Lemma 3.2, take a sequence $(\beta_{n})\subset(0,+\infty)$ such
that $(\beta_{n}f_{n})$ is Bessel with bound $B=A^{-1}\|T^{-1}\|^{-2}$.
Similar to the above paragraph, using the Cauchy-Schwarz inequality, we get
that the sequence $(\beta_{n}^{-1}h_{n})$ is a lower semi frame for ${\cal H}$
with bound $B^{-1}$. Then $(\beta_{n}^{-1}g_{n})$ is a lower semi frame for
${\cal H}$ with bound $B^{-1}\|T^{-1}\|^{-2}=A$.

Since $(\lambda_{n}^{-1}f_{n})$ and $(\beta_{n}^{-1}g_{n})$ are complete in
${\cal H}$, clearly $(\lambda_{n}g_{n})$ and $(\beta_{n}f_{n})$ are also
complete in ${\cal H}$. ∎

As particular case, Proposition 3.3 applies to sequences $(f_{n})$ and
$(g_{n})$ which are weakly dual to each other in the sense that

$\langle f,g\rangle=\sum_{n}\langle f,f_{n}\rangle\langle
g_{n},g\rangle,\qquad\forall f,g\in{\cal H},$ (4)

or when the operator $M$ given by

$Mf=\sum_{n}\langle f,g_{n}\rangle f_{n}$ (5)

is well defined on ${\cal H}$ and belongs to $GL({\cal H})$. Note that an
operator $M$ determined by (5) is a multiplier with a constant symbol (see
Section 4.4 for the definition of multipliers); see [36, 37, 42] for results
related to the question when a multiplier is in $GL({\cal H})$. For studies of
representations of unbounded operators like (5) we refer to [12].

In particular, by the proofs of Proposition 3.3 and Lemma 3.2, the following
can be stated for the important case of weakly dual sequences:

######  Corollary 3.4.

Let $((f_{n}),(g_{n}))$ be a weakly dual pair for ${\cal H}$, i.e., (4) holds.
Then there are weights $(\lambda_{n})\subset(0,\infty)$ and
$(\beta_{n})\subset(0,\infty)$ such that
$(\frac{1}{\lambda_{n}}f_{n},\lambda_{n}g_{n})$ and
$(\beta_{n}f_{n},\frac{1}{\beta_{n}}g_{n})$ are weakly dual pairs for ${\cal
H}$, $(\frac{1}{\lambda_{n}}f_{n})$ and $(\frac{1}{\beta_{n}}g_{n})$ are lower
semi-frames for ${\cal H}$, and $(\lambda_{n}g_{n})$ and $(\beta_{n}f_{n})$
are upper semi-frames for ${\cal H}$. The weights can be chosen e.g. as
$\lambda_{n}=\tau_{n}\|g_{n}\|^{-1}$ if $g_{n}\neq 0$, $\lambda_{n}=1$ if
$g_{n}=0$, $\beta_{n}=\tau_{n}\|f_{n}\|^{-1}$ if $f_{n}\neq 0$, and
$\beta_{n}=1$ if $f_{n}=0$, where $(\tau_{n})\subset(0,\infty)$ is a sequence
such that $\sum_{n}\tau_{n}^{2}=1$.

As another direct application of Proposition 3.3 we have the following
statement for Schauder bases333Recall that a sequence $(f_{n})$ in ${\cal H}$
is a Schauder basis for ${\cal H}$ if there exists a unique sequence $(g_{n})$
in ${\cal H}$ such that $f=\sum_{n}\langle f,g_{n}\rangle f_{n}$ for every
$f\in{\cal H}$..

######  Corollary 3.5.

Let $(f_{n})$ be a sequence in ${\cal H}$ containing a Schauder basis for
${\cal H}$ or a frame for ${\cal H}$. Then Conjecture 3.1 is true for
$(f_{n})$.

Recall that when $(f_{n})$ is a sequence in ${\cal H}$ whose synthesis
operator is closed and surjective, then there is a Bessel sequence $(g_{n})$
such that $f=\sum\langle f,g_{n}\rangle f_{n}$ for every $f\in{\cal H}$ and
$(f_{n})$ satisfies the lower frame condition for ${\cal H}$, see [17, Theorem
8.4.1]. Let us now consider the following (more general) case using closed and
surjective operator of the form (5):

######  Proposition 3.6.

Let $(f_{n})$ and $(g_{n})$ be sequences in ${\cal H}$. Define the operator
$M$ by (5) with domain determined by those $f$ for which the series in (5)
converges. If $M$ is closed and $R(M)={\cal H}$, then Conjecture 3.1 is true
for $(f_{n})$.

###### Proof.

By [13, Lemma 1.1, Corollary 1.2], the operator $M$ admits a pseudo-inverse
$M^{+}$ which belongs to $\mathcal{B}({\cal H})$ since $R(M)={\cal H}$. This
means, in particular, that $MM^{+}h=h$ for every $h\in{\cal H}$. Then

$h=MM^{+}h=\sum_{n}\langle M^{+}h,g_{n}\rangle f_{n}=\sum_{n}\langle
h,M^{+*}g_{n}\rangle f_{n},\qquad\forall h\in{\cal H}.$

By Corollary 3.4, Conjecture 3.1 is true for $(f_{n})$. ∎

We proceed with one more class of sequences for which Conjecture 3.1 holds.

######  Proposition 3.7.

Any complete sequence in ${\cal H}$ whose analysis operator has finite
dimensional domain is a lower semi frame for ${\cal H}$ (thus, Conjecture 3.1
trivially holds for it).

###### Proof.

Let $(f_{n})$ be a complete sequence in ${\cal H}$ such that its analysis
operator $C$ has finite dimensional (closed) domain ${\cal D}(C)$. Then for
$f\in{\cal D}(C)$ we have

$\sum_{n=1}^{\infty}|\langle f,f_{n}\rangle|^{2}=\sum_{n=1}^{\infty}|\langle
f,Pf_{n}\rangle|^{2},$ (6)

where $P$ is the orthogonal projection onto ${\cal D}(C)$. The sequence
$(Pf_{n})$ is complete in ${\cal D}(C)$ and since $\dim{\cal D}(C)=d<\infty$,
there exist $f_{n_{1}},f_{n_{2}},\dots,f_{n_{d}}$ such that
$Pf_{n_{1}},Pf_{n_{2}},\dots,Pf_{n_{d}}$ are linearly independent and thus a
Riesz basis for ${\cal D}(C)$. This implies that $(Pf_{n})$ is a lower semi
frame for ${\cal D}(C)$. By (6), $(f_{n})$ satisfies the lower frame condition
for ${\cal D}(C)$ and hence $(f_{n})$ is a lower semi frame for ${\cal H}$. ∎

Note that if the analysis operator $C$ of $(f_{n})$ has finite dimensional
domain then also the orthogonal complement of $(f_{n})$ is finite dimensional.
This follows by the fact that if $h\perp f_{n}$ for every $n$, then $h\in{\cal
D}(C)$.

######  Remark 3.8.

As it can be expected, the above propositions do not cover all the possible
classes of sequences for which Conjecture 3.1 holds. As an illustration,
consider for example the sequence
$(f_{n})_{n=2}^{\infty}=(n(e_{1}+e_{n}))_{n=2}^{\infty}$ where
$(e_{n})_{n=1}^{\infty}$ denotes an orthonormal basis for ${\cal H}$. It is
shown in [33] that $(f_{n})_{n=2}^{\infty}$ satisfies the lower frame
condition for ${\cal H}$, but not all the elements of ${\cal H}$ can be
represented in the form $\sum_{n}c_{n}f_{n}$ (hence, Proposition 3.6 does not
apply) and the representation (2) can not be extended to the entire space
${\cal H}$. In a similar way as in [33] it can be shown that $(f_{n})$ does
not generate a reproducing pair $(f_{n},g_{n})$ for ${\cal H}$ and thus
Proposition 3.3 does not apply. Clearly, the domain of the analysis operator
of $(f_{n})$ is not finite dimensional, so Proposition 3.7 does not apply
either.

The above remark motivates interest to further consideration of Conjecture
3.1. Let us end this section showing that we can reduce the conjecture
studying only sequences with densely defined analysis operators.

######  Theorem 3.9.

The following statements are equivalent.

  1. 1.

Conjecture 3.1 is true.

  2. 2.

Conjecture 3.1 is true for every complete sequence in ${\cal H}$ whose
analysis operator is densely defined 444i.e., whose synthesis operator is
closable [11]; in such case one also has $D\subseteq C^{*}$ [16]..

###### Proof.

$1.\implies 2.$ Clear.
$2.\implies 1.$ Let $\mathcal{F}:=(f_{n})$ be a complete sequence in ${\cal
H}$ with analysis operator $C_{\mathcal{F}}$. We will prove that Conjecture
3.1 holds for $F$ based on the assumption for validity of $2.$ Let
$W=\overline{{\cal D}(C_{\mathcal{F}})}$ be the closure of ${\cal
D}(C_{\mathcal{F}})$ in ${\cal H}$ and let $P_{W}$ be the orthogonal
projection onto $W$. If $\dim W<\infty$, then the conjecture is true for
$\mathcal{F}$ by Proposition 3.7. Assume that $W$ has infinite dimension.
Recall that ${\cal H}$ is separable, therefore $W$ is separable and there
exists a unitary operator $J:W\to{\cal H}$ (i.e., $J^{*}=J^{-1}$). Define
$(h_{n}):=(P_{W}f_{n})$ and $\mathcal{G}:=(g_{n}):=(Jh_{n})$. The sequences
$(h_{n})$ and $(g_{n})$ are complete in $W$ and ${\cal H}$, respectively. Let
$C_{\mathcal{G}}:{\cal D}(C_{\mathcal{G}})\subset{\cal H}\to\ell^{2}$ be the
analysis operator of $\mathcal{G}$. Observe that for $f\in W$ the following
relations hold

$\sum_{n}|\langle Jf,g_{n}\rangle|^{2}=\sum_{n}|\langle
f,h_{n}\rangle|^{2}=\sum_{n}|\langle P_{W}f,f_{n}\rangle|^{2}=\sum_{n}|\langle
f,f_{n}\rangle|^{2},$

which implies that $Jf\in{\cal D}(C_{\mathcal{G}})$ if and only if $f\in{\cal
D}(C_{\mathcal{F}})$. Therefore, $J{\cal D}(C_{\mathcal{F}})={\cal
D}(C_{\mathcal{G}})$, leading to the conclusion that ${\cal
D}(C_{\mathcal{G}})$ is dense in ${\cal H}$.

Fix an arbitrary $A>0$. Since $(g_{n})$ is complete in ${\cal H}$ with ${\cal
D}(C_{\mathcal{G}})$ being dense in ${\cal H}$, by assumption there is a
sequence $(\omega_{n})\subset(0,\infty)$ such that $(\omega_{n}g_{n})$ is a
lower semi frame for ${\cal H}$ with bound $A$. By making the substitution
$\omega_{n}\to\omega_{n}+1$, we can assume that $\omega_{n}>1$ for every $n$.
It remains to prove that $(\omega_{n}f_{n})$ is a lower semi frame for ${\cal
H}$ with bound $A$.

Let $f\in{\cal H}$. If $\sum_{n}|\langle
f,\omega_{n}f_{n}\rangle|^{2}=\infty$, then the lower frame inequality
trivially holds. Now consider the remaining case. In this case we have

$\sum_{n}|\langle f,f_{n}\rangle|^{2}\leq\sum_{n}|\langle
f,\omega_{n}f_{n}\rangle|^{2}<\infty,$

so $f$ is necessarily in ${\cal D}(C_{\mathcal{F}})\subset W$. Therefore,

$\sum_{n}|\langle f,\omega_{n}f_{n}\rangle|^{2}=\sum_{n}|\langle
P_{W}f,\omega_{n}f_{n}\rangle|^{2}=\sum_{n}|\langle
f,\omega_{n}h_{n}\rangle|^{2}=\sum_{n}|\langle
Jf,\omega_{n}g_{n}\rangle|^{2}\geq A\|Jf\|^{2}=A\|f\|^{2}.\qed$

In [8] a similar approach looking at the closure of the domain of the analysis
operator is used for results of lower frames sequences and Riesz-Fisher
sequences.

##  4 Weighted frames

Note that in the literature [29, 14, 6] Definition 1.1 was used to define the
concept of $w$-frame while weighted frames were used either with the meaning
of a $w$-frame or with the meaning of a sequence which can be transformed to a
frame via a weight $(w_{n})\subset\mathbb{C}$. Trivially, Definition 1.1 is
equivalent to a definition which requires $(\omega_{n})\subset\mathbb{C}$ and
$\omega_{n}\neq 0$ for every $n$ (taking $(|\omega_{n}|)$ in consideration).
Notice that Lemma 3.2 can bring conclusion for equivalence of Definition 1.1
to a definition that furthermore allows zero elements in $(w_{n})$ (so, one
can also use $[0,+\infty]$ or the entire set $\mathbb{C}$ for an equivalent
definition). Indeed, assume that $(f_{n})$ is a sequence such that
$(\omega_{n}^{\prime}f_{n})$ is a frame for ${\cal H}$ for some
$(\omega_{n}^{\prime})\subset[0,\infty)$. Denote
$I=\\{n\in\mathbb{N}:\omega_{n}^{\prime}\neq 0\\}$ and split
$(f_{n})=(f_{m})_{m\in I}\cup(f_{k})_{k\notin I}$. By Lemma 3.2, there exists
a sequence $(\tau_{k})_{k\notin I}\subset(0,\infty)$ such that
$(\tau_{k}f_{k})_{k\notin I}$ is a Bessel sequence. Now, define
$\omega_{n}=\omega_{n}^{\prime}$ if $n\in I$ and $\omega_{n}=\tau_{n}$ if
$n\notin I$. By construction, $(\omega_{n})\subset(0,+\infty)$. Since
$(\omega_{m}^{\prime}f_{m})_{m\in I}$ is a frame for ${\cal H}$ and
$(\tau_{k}f_{k})_{k\notin I}$ is a Bessel sequence, the sequence
$(\omega_{n}f_{n})$ is a frame for ${\cal H}$.

###  4.1 Necessary conditions

We begin with some necessary conditions that are easy consequences of known
results about frames:

######  Corollary 4.1.

If $(f_{n})$ is a weighted frame for ${\cal H}$, then $(f_{n})$ is complete in
${\cal H}$ and there exists a sequence $(g_{n})$ in ${\cal H}$ such that

$f=\sum_{n}\langle f,f_{n}\rangle g_{n}=\sum_{n}\langle f,g_{n}\rangle
f_{n},\qquad\forall f\in{\cal H},$

with unconditional convergence.

As in the case with frames, where completeness is necessary and not sufficient
condition when the space is infinite dimensional, here completeness is also
necessary and not sufficient for the weighted frame property. An illustration
is given in the following example.

######  Example 4.2.

Let $(f_{n})_{n=2}^{\infty}=(e_{1}+e_{n})_{n=2}^{\infty}$, where
$(e_{n})_{n=1}^{\infty}$ is an orthonormal basis for ${\cal H}$. It is easy to
see that $(f_{n})_{n=2}^{\infty}$ is complete in ${\cal H}$. However,
$(f_{n})_{n=2}^{\infty}$ is not a weighted frame for ${\cal H}$. Indeed, with
an argument similar to the one used in [33], there does not exist a sequence
$(g_{n})_{n=2}^{\infty}$ such that

$f=\sum_{n}\langle f,f_{n}\rangle g_{n},\qquad\forall f\in{\cal H}.$

Then, by Proposition 4.1, $(f_{n})_{n=2}^{\infty}$ is not a weighted frame for
${\cal H}$.

The next statement gives a class of complete sequences that are not weighted
frames. For recent results on completeness in relation to the Bessel and Riesz
basis properties we reefer to [35].

######  Proposition 4.3.

Let $(f_{n})$ be a complete minimal sequence in ${\cal H}$ whose biorthogonal
sequence is not complete in ${\cal H}$. Then $(f_{n})$ is not a weighted frame
for ${\cal H}$.

###### Proof.

Suppose that $(f_{n})$ is a weighted frame for ${\cal H}$ and a minimal
sequence. In particular, this means that there exists
$(\omega_{n})\subset(0,+\infty)$ such that $(\omega_{n}f_{n})$ is a Riesz
basis for ${\cal H}$. Then the biorthogonal sequence to $(\omega_{n}f_{n})$ is
a Riesz basis for ${\cal H}$ too. This implies that the biorthogonal sequence
to $(f_{n})$ is complete in ${\cal H}$, which leads to a contradiction. ∎

######  Remark 4.4.

The sequence $(f_{n})_{n=2}^{\infty}$ in Example 4.2 is minimal and complete
in ${\cal H}$, and its biorthogonal sequence is $(e_{n})_{n=2}^{\infty}$,
which is not complete in ${\cal H}$. Hence, Proposition 4.3 gives another
proof that $(f_{n})_{n=2}^{\infty}$ is not a weighted frame for ${\cal H}$.

Further necessary conditions for the weighted frame property are given in the
next proposition:

######  Proposition 4.5.

Let $(f_{n})$ be a sequence in ${\cal H}$ and $(\omega_{n})\subset(0,+\infty)$
be such that $(\omega_{n}f_{n})$ is a Bessel sequence in ${\cal H}$. Then

  1. 1.

$\sup_{n}(\omega_{n}\|f_{n}\|)<\infty$;

  2. 2.

if $\inf_{n}\|f_{n}\|>0$, then $\sup_{n}(\omega_{n})<\infty$;

  3. 3.

if $(f_{n})$ is not a Bessel sequence, then $\inf_{n}(\omega_{n})=0$.

###### Proof.

  1. Statement 1. comes from a well known result that the elements of a Bessel sequence are norm-bounded from above. Statement 2. follows trivially from 1.

  2. 3.

Assume that $(f_{n})$ is not a Bessel sequence and $\inf_{n}(\omega_{n})=c>0$.
In this case there exists $f\in{\cal H}$ such that $\sum_{n}|\langle
f,f_{n}\rangle|^{2}=+\infty,$ which leads to a contradiction:

$\infty=c^{2}\sum_{n}|\langle f,f_{n}\rangle|^{2}\leq\sum_{n}|\langle
f,\omega_{n}f_{n}\rangle|^{2}<\infty.\qed$

######  Remark 4.6.

Let us come back again to the sequence $(f_{n})_{n=2}^{\infty}$ from Example
4.2. As it was proved above in different ways, $(f_{n})_{n=2}^{\infty}$ is not
a weighted frame for ${\cal H}$. Here, we give another proof by applying
Proposition 4.5. Assume that $(\omega_{n}f_{n})_{n=2}^{\infty}$ is a frame for
${\cal H}$ for some $(\omega_{n})\subset(0,+\infty)$. Since
$(f_{n})_{n=2}^{\infty}$ is not a Bessel sequence,
$\inf_{n}(\omega_{n})_{n=2}^{\infty}=0$ by Proposition 4.5, and thus there
exists a subsequence $(\omega_{n_{k}})$ such that
$\displaystyle\lim_{k\to+\infty}\omega_{n_{k}}=0$. This leads to the following
contradiction:

$1=\|e_{n_{k}}\|^{2}\approx\sum_{n}\omega_{n}^{2}|\langle
e_{n_{k}},f_{n}\rangle|^{2}=\omega_{n_{k}}^{2}\to 0\mbox{ as $k\to\infty$}.$

###  4.2 Sufficient conditions

######  Proposition 4.7.

If a sequence $(f_{n})$ contains a proper subsequence which is a weighted
frame for ${\cal H}$, then $(f_{n})$ is a weighted frame for ${\cal H}$.

###### Proof.

Let us assume that $(f_{n})=(f_{n_{k}})_{k\in I}\cup(g_{m})_{m\in J}$ with
some (not necessarily infinite) sequence $(g_{m})_{m\in J}$ in ${\cal H}$ such
that $(\omega_{k}f_{n_{k}})_{k\in I}$ is a frame for ${\cal H}$ with
$(\omega_{k})\subset(0,+\infty)$. Then, by Lemma 3.2, there exists a sequence
$(\lambda_{m})_{m\in J}\subset(0,+\infty)$ such that $(\lambda_{m}g_{m})_{m\in
J}$ is a Bessel sequence. Hence, $(\omega_{k}f_{n_{k}})_{k\in
I}\cup(\lambda_{m}g_{m})_{m\in J}$ is a frame for ${\cal H}$ and $(f_{n})$ is
a weighted frame for ${\cal H}$. ∎

We recall that the excess of a sequence $(f_{n})$ in ${\cal H}$ is defined as

$e((f_{n}))=\sup\\{|J|:J\subseteq I\text{ and
}\overline{\text{span}}\\{f_{n}\\}_{n\in I\backslash
J}=\overline{\text{span}}\\{f_{n}\\}_{n\in I}\\},$

and, moreover, a frame has null excess if and only if it is a Riesz basis
(see, e.g., [26, Section 8.7])555The excess of frames have been studied also
in [4, 3, 27, 28].. The next statement gives a description of the class of
weighted frames among all sequences with finite positive access.

######  Proposition 4.8.

If a sequence in ${\cal H}$ has finite positive excess, then it is a weighted
frame for ${\cal H}$ if and only if it contains a proper subsequence which is
a weighted frame for ${\cal H}$ if and only if it contains a proper
subsequence which is a weighted Riesz basis for ${\cal H}$.

###### Proof.

One of the directions follows from Proposition 4.7. For the other direction,
assume that $(f_{n})$ is a weighted frame for ${\cal H}$ with finite positive
excess. Then there exists $(\omega_{n})\subset(0,+\infty)$ such that
$(\omega_{n}f_{n})$ is a frame for ${\cal H}$. Clearly, $(\omega_{n}f_{n})$
has finite positive excess. Hence, by [26, Lemma 8.42], there exists a proper
subsequence of $(\omega_{n}f_{n})$ which is a Riesz basis for ${\cal H}$.
Thus, $(f_{n})$ contains a weighted Riesz basis. ∎

###  4.3 Some characterizations of weighted frames

Here, we consider more general operators than the analysis and synthesis ones.
Given a sequence $\mathcal{F}=(f_{n})$ in ${\cal H}$, we consider the operator
$\mathsf{C}_{\mathcal{F}}:{\cal H}\to\mathbb{C}^{\mathbb{N}}$ defined by
$\mathsf{C}_{\mathcal{F}}f=(\langle f,f_{n}\rangle)$, and the operator
$\mathsf{D}_{\mathcal{F}}:{\cal
D}(\mathsf{D}_{\mathcal{F}})\subseteq\mathbb{C}^{\mathbb{N}}\to{\cal H}$
determined by $\mathsf{D}_{\mathcal{F}}(c_{n})=\sum_{n}c_{n}f_{n}$ for those
complex sequences $(c_{n})$ for which $\sum_{n}c_{n}f_{n}$ converges in ${\cal
H}$. Note that $\mathsf{C}_{\mathcal{F}}$ is a linear operator between vector
spaces and it is equal to the analysis operator of $(f_{n})$ if and only if
$(f_{n})$ is a Bessel sequence. The following characterizations of the
weighted frame property are inspired and based on known characterizations of
the frame property [17, Corollary 5.5.3 and Theorem 5.5.1].

######  Proposition 4.9.

Given a sequence $\mathcal{F}=(f_{n})$ in ${\cal H}$, the following statements
are equivalent.

1\. $(f_{n})$ is a weighted frame for ${\cal H}$.

2\. There exists $(\omega_{n})\subset(0,+\infty)$ such that
$R(\mathsf{C}_{\mathcal{F}})$ is a Hilbert space with inner product
$\langle\cdot,\cdot\rangle_{\omega^{2}}$ defined by

$\langle(c_{n}),(d_{n})\rangle_{\omega^{2}}=\sum_{n}\omega_{n}^{2}c_{n}\overline{d_{n}},$

and $\mathsf{C}_{\mathcal{F}}$ is a bounded and bijective operator between
${\cal H}$ and $R(\mathsf{C}_{\mathcal{F}})$.

3\. There exists $(\omega_{n})\subset(0,+\infty)$ such that the operator
$\mathsf{D}_{\mathcal{F}}$ is well-defined on the Hilbert space
$\ell^{2}_{\omega^{-2}}$ determined by

$\ell^{2}_{\omega^{-2}}=\left\\{(c_{n})\in\mathbb{C}^{\mathbb{N}}:\sum_{n}\frac{|c_{n}|^{2}}{\omega_{n}^{2}}<\infty\right\\},\
\ \
\langle(c_{n}),(d_{n})\rangle_{\omega^{-2}}=\sum_{n}\frac{c_{n}\overline{d_{n}}}{\omega_{n}^{2}},$

and $R(\mathsf{D_{\mathcal{F}}})={\cal H}$.

###  4.4 A conjecture about weighted frames and relation to unconditionally
convergent multipliers

Proposition 4.1 suggests the following conjecture:

######  Conjecture 4.10.

A sequence $(f_{n})$ in ${\cal H}$ is a weighted frame for ${\cal H}$ if and
only if there is a sequence $(g_{n})$ so that for every $f\in{\cal H}$ one has
$f=\sum_{n}\langle f,f_{n}\rangle g_{n}$ with unconditional convergence.

First note that the representation $f=\sum_{n}\langle f,f_{n}\rangle g_{n}$ in
the above conjecture can equivalently be replaced by $f=\sum_{n}\langle
f,g_{n}\rangle f_{n}$, because validity of $f=\sum_{n}\langle f,f_{n}\rangle
g_{n}$ with unconditional convergence for every $f\in{\cal H}$ is equivalent
to validity of $f=\sum_{n}\langle f,g_{n}\rangle f_{n}$ with unconditional
convergence for every $f\in{\cal H}$ (see [38, Lemmas 3.1 and 2.3]).

As it was noticed in Prop. 4.1, the “only if” direction of Conjecture 4.10
holds true, so only the “if” direction is still open. The “if” direction can
be related to conjectures and results on invertible unconditionally convergent
multipliers in [38]. Let us first recall that a multiplier of two sequences
$\Phi=(\varphi_{n})$ and $\Psi=(\psi_{n})$ in ${\cal H}$ and a symbol (complex
sequence) $m=(m_{n})$ is the operator $M_{m,\Phi,\Psi}$ given by

$M_{m,\Phi,\Psi}f=\sum_{n}m_{n}\langle f,\psi_{n}\rangle\varphi_{n}$ (7)

for those $f$ from ${\cal H}$ for which the series in (7) converges.
Multipliers were extensively studied in [5, 10, 20, 21, 23, 36, 37, 38, 39,
40, 41, 42] and generalized in different ways [7, 19]. Note that a multiplier
which is defined on the entire space ${\cal H}$ is automatically bounded on
${\cal H}$ [38]. A multiplier $M_{m,\Phi,\Psi}$ is said to be unconditionally
convergent if the series in (7) is unconditionally convergent for every
$f\in{\cal H}$. The multiplier of two Bessel sequences and bounded symbol
belongs to $\mathcal{B}({\cal H})$ and it is unconditionally convergent [5].
In [38], a main topic was a converse question, namely, the question whether an
unconditionally convergent multiplier can be written as a multiplier of two
Bessel sequences and symbol (1) just by shifting of weights. This question was
affirmatively answered for some classes of multipliers [38, 25] and the
general case was posed as a conjecture:

######  Conjecture (M1) ([38]).

Let $M_{m,\Phi,\Psi}$ be an unconditionally convergent multiplier. Then there
exist $(\alpha_{n})\subset\mathbb{C}$ and $(\beta_{n})\subset\mathbb{C}$ such
that $(\alpha_{n}\varphi_{n})$ and $(\beta_{n}\psi_{n})$ are Bessel sequences
and $\alpha_{n}\overline{\beta_{n}}=m_{n}$ for all $n$.

A question in a similar spirit, related to representation of an invertible
multiplier as a multiplier of frames by shifting of weights, was also
investigated in [38] and it can be stated as a conjecture too:

######  Conjecture (M2) ([38]).

Let $M_{m,\Phi,\Psi}$ be an unconditionally convergent multiplier which is
invertible on ${\cal H}$. Then there exist $(\alpha_{n})\subset\mathbb{C}$ and
$(\beta_{n})\subset\mathbb{C}$ such that $(\alpha_{n}\varphi_{n})$ and
$(\beta_{n}\psi_{n})$ are frames for ${\cal H}$ and
$\alpha_{n}\overline{\beta_{n}}=m_{n}$ for all $n$.

The two conjectures (M1) and (M2) are actually related. That validity of (M1)
implies validity of (M2) was observed already in [38]. Surprisingly, the
converse implication turned out to also be true.

######  Theorem 4.11.

Conjectures (M1) and (M2) are equivalent.

###### Proof.

(M1) $\implies$ (M2) See [38, Section 4] for a proof.

(M2) $\implies$ (M1) Let $M_{m,\Phi,\Psi}$ be an unconditionally convergent
multiplier and thus a bounded operator on ${\cal H}$ as well. Write
$I-M_{m,\Phi,\Psi}=M_{(1),\mathcal{F},\mathcal{G}}$ for some multiplier
$M_{(1),\mathcal{F},\mathcal{G}}$ with symbol $(1)$ and Bessel sequences
$\mathcal{F}=(f_{n})$ and $\mathcal{G}=(g_{n})$ (for instance, take
$\mathcal{G}$ to be an orthonormal basis for ${\cal H}$ and
$f_{n}=(I-M_{m,\Phi,\Psi})g_{n}$ for every $n$). Define
$\Xi=(\xi_{n})=(\varphi_{1},f_{1},\varphi_{2},f_{2},\dots)$,
$\Theta=(\theta_{n})=(\psi_{1},g_{1},\psi_{2},g_{2},\dots)$, and
$m^{\prime}=(m_{n}^{\prime})=(m_{1},1,m_{2},1,\dots)$. It is easy to see that
$M_{m^{\prime},\Xi,\Theta}$ is unconditionally convergent and equals to $I$.
Then, if Conjecture (M2) is true, there exist $(\alpha_{n})\subset\mathbb{C}$
and $(\beta_{n})\subset\mathbb{C}$ such that $(\alpha_{n}\xi_{n})$ and
$(\beta_{n}\theta_{n})$ are frames for ${\cal H}$ and
$\alpha_{n}\overline{\beta_{n}}=m_{n}^{\prime}$ for every $n$, so
$(\alpha_{2n-1}\varphi_{n})$ and $(\beta_{2n-1}\psi_{n})$ are Bessel sequences
and $\alpha_{2n-1}\overline{\beta_{2n-1}}=m_{2n-1}^{\prime}=m_{n}$ for all
$n$. ∎

Let us now return to Conjecture 4.10 and its relation to the above
conjectures. Using the equivalence of definitions of weighted frames discussed
at the beginning of Section 4, it is clear that validity of Conjecture (M2)
implies validity of the “if”-direction of Conjecture 4.10 and thus one can
state the following:

######  Proposition 4.12.

Conjecture (M2) implies Conjecture 4.10.

It is still an open question whether the converse holds, i.e., whether
validity of Conjecture 4.10 implies validity of Conjecture (M2).

We end the section with some classes of sequences for which Conjecture 4.10 is
true.

######  Proposition 4.13.

Let $(f_{n})$ and $(g_{n})$ be sequences in ${\cal H}$ such that
$\inf_{n}\|f_{n}\|\|g_{n}\|>0$ or $(g_{n})$ is minimal or $(f_{n})$ is
minimal. The following statements are equivalent.

1\. For every $f\in{\cal H}$, $f=\sum\limits_{n}\langle f,f_{n}\rangle g_{n}$
with unconditional convergence.

2\. $(\|g_{n}\|f_{n})$ and $(\|g_{n}\|^{-1}g_{n})$ are dual frames for ${\cal
H}$.

3\. $(\|f_{n}\|g_{n})$ and $(\|f_{n}\|^{-1}f_{n})$ are dual frames for ${\cal
H}$.

###### Proof.

The implication $2.\Rightarrow 1.$ (resp. $3.\Rightarrow 1.$) follows from
well known results in frame theory for any two sequences $(f_{n})$ and
$(g_{n})$ with $g_{n}\neq 0$ for every $n$ (resp. $(g_{n})$ and $(f_{n})$ with
$f_{n}\neq 0$ for every $n$).

For the converse implications, assume that 1. holds. If
$\inf_{n}\|f_{n}\|\|g_{n}\|>0$, then [38, Corollary 3.6] implies that
$(\|g_{n}\|f_{n})$ and $(\|g_{n}\|^{-1}g_{n})$ are Bessel sequences and then
[17, Lemma 6.3.2] completes the proof that 2. holds. If $(g_{n})$ is minimal,
then the proof of [38, Proposition 4.8] implies that
$\inf_{n}\|f_{n}\|\|g_{n}\|>0$ and thus 2. holds by what is already proved. As
mentioned at the beginning of Section 4.4, statement 1. is equivalent to
validity of $f=\sum_{n}\langle f,g_{n}\rangle f_{n}$ with unconditional
convergence for every $f\in{\cal H}$. Thus, 3. holds again from what is
already proved. Finally, if $(f_{n})$ is minimal, then 2. and 3. hold with
similar arguments. ∎

## Acknowledgments

The first author thanks M. Speckbacher and D. Freeman for related discussions
in the course of preparing “Quantitative bounds for unconditional pairs of
frames”, P. Balazs, D. Freeman, R. Popescu, M. Speckbacher, arXiv:2212.00947,
and acknowledges the support by the project P 34624 “Localized, Fusion and
Tensors of Frames” (LoFT) of the Austrian Science Fund (FWF). The second
author thanks P. Balazs and D. Stoeva for the hospitality at the Acoustics
Research Institute in 2019 and acknowledges partial supports by the Università
degli Studi di Palermo (through FFR2023 “Corso”) and by the “Gruppo Nazionale
per l’Analisi Matematica, la Probabilità e le loro Applicazioni” (INdAM). The
third author acknowledges support from the Austrian Science Fund (FWF) through
Project P 35846-N “Challenges in Frame Multiplier Theory”.

## References

  * [1] J.-P. Antoine, P. Balazs, Frames and semi-frames, J. Phys. A: Math. Theor. 44, 205201 (2011); Corrigendum 44, (2011), 479501.
  * [2] J.-P. Antoine, P. Balazs, Frames, semi-frames, and Hilbert scales, Numer. Funct. Anal. Optim. 33, (2012), 736–769.
  * [3] D. Bakić, T. Berić, On excesses of frames, Glas. Mat., 50, (2015), 415–427.
  * [4] R. Balan, P. Casazza, C. Heil, Z. Landau, Deficits and excesses of frames, Adv. Comput. Math. 18, (2003), 93–116.
  * [5] P. Balazs, Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl., 325 (1), (2007), 571–585.
  * [6] P. Balazs, J.-P. Antoine, A. Grybos.  Weighted and controlled frames: Mutual relationship and first numerical properties.  Int. J. Wavelets Multiresolut. Inf. Process., 8(1), (2010), 109–132.
  * [7] P. Balazs, D. Bayer, A. Rahimi, Multipliers for continuous frames in Hilbert spaces, J. Phys. A: Math. Theor., 45(24), (2012), 244023.
  * [8] P. Balazs, M. Shamsabadi, An unbounded operator theory approach to lower frame sequence and Riesz-Fisher sequences, preprint, https://arxiv.org/abs/2310.02088.
  * [9] P. Balazs, M. Speckbacher, Reproducing pairs and Gabor systems at critical density, J. Math. Anal. Appl. 455(2), (2017), 1072–1087.
  * [10] P. Balazs, D. T. Stoeva, Representation of the inverse of a frame multiplier, J. Math. Anal. Appl., 422(2), (2015), 981–994.
  * [11] P. Balazs, D. T. Stoeva, J.-P. Antoine, Classification of general sequences by frame related operators, Sampl. Theory Signal Image Process. 10, (2011), 151–170.
  * [12] G. Bellomonte, R. Corso, Frames and weak frames for unbounded operators, Adv. Comput. Math., 46(2), 38, (2020), 21pp.
  * [13] F. J. Beutler, W. L. Root, The operator pseudo-inverse in control and systems identifications, In: Generalized Inverses and Applications, M. Z. Nashed (ed.), Academic Press, 1976, 397–494.
  * [14] I. Bogdanova, P. Vandergheynst, J.-P. Antoine, L. Jacques, M. Morvidone, Stereographic wavelet frames on the sphere, Appl. Comput. Harmon. Anal., 19, (2005), 223–252.
  * [15] P.G. Casazza, O. Christensen, S. Li, A. Linder. Riesz-Fischer Sequences and Lower Frame Bounds, Z. Anal. Anwend. 21(2), (2002), 305–314.
  * [16] O. Christensen, Frames and Pseudo-inverses, J. Math. Anal. Appl. 195, (1995), 401–414.
  * [17] O. Christensen,  An Introduction to Frames and Riesz Bases. Second edition.  Birkhäuser, 2016.
  * [18] R. Corso, Sesquilinear forms associated to sequences on Hilbert spaces, Monatsh. Math., 189(4), (2019), 625–650.
  * [19] R. Corso, F. Tschinke, Some notes about distribution frame multipliers, in: Landscapes of Time-Frequency Analysis, vol. 2, P. Boggiatto, T. Bruno, E. Cordero, H.G. Feichtinger, F. Nicola, A. Oliaro, A. Tabacco, M. Vallarino (Ed.), Appl. Numer. Harmon. Anal. Series, Springer, 2020.
  * [20] R. Corso, On some dual frames multipliers with at most countable spectra, Ann. Mat. Pura Appl., 201(4), (2022), 1705–1716.
  * [21] R. Corso, Localization of the spectra of dual frames multipliers, Constructive Mathematical Analysis, 5(4), (2022), 238–245.
  * [22] R. Corso, Generalized frame operator, lower semi-frames and sequences of translates, Math. Nachr., 296(7), (2023), 2715–2733.
  * [23] R. Corso, Estimate of the spectral radii of Bessel multipliers and consequences, Constructive Mathematical Analysis, 6(3), (2023), 176–183.
  * [24] J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366.
  * [25] C. Fernández, A. Galbis, E. Primo, Unconditionally convergent multipliers and Bessel sequences, J. Math. Anal. Appl. 455(1), (2017), 389–395.
  * [26] C. Heil, A Basis Theory Primer, Expanded Edition, Birkhäuser, 2010.
  * [27] C. Heil, P.T. Yu, Convergence of Frame Series, J. Fourier Anal. Appl., 29, (2023), 14.
  * [28] J. Holub, Pre-frame operators, Besselian frames and near-Riesz bases, Proc. Am. Math. Soc. 122, (1994), 779–785.
  * [29] L. Jacques, Ondelettes, repères et couronne solaire, Thèse de Doctorat, Univ. Cath. Louvain, Louvain-la-Neuve, 2004.
  * [30] T. Kato, Perturbation theory for linear operators, Springer Berlin, Heidelberg, 1966.
  * [31] G. Kutyniok, K.A. Okoudjou, F. Phillip, E.K. Tuley, Scalable frames, Linear Algebra Appl. 438(5), (2013), 2225-2238.
  * [32] M. Speckbacher, P. Balazs.  Reproducing pairs and the continuous nonstationary Gabor transform on lca groups.  J. Phys. A, (2015), 48:395201.
  * [33] D. T. Stoeva. Connection between the lower p-frame condition and existence of reconstruction formulas in a Banach space and its dual. Ann. Sof. Univ., Fac. Math. and Inf. 97, (2005), 123-133.
  * [34] D. T. Stoeva. Characterization of atomic decompositions, Banach frames, Xd-frames, duals and synthesis-pseudo-duals, with application to Hilbert frame theory. arXiv:1108.6282
  * [35] D. T. Stoeva. On a Characterization of Riesz Bases via Biorthogonal Sequences. J. Fourier Anal. Appl. 26, (2020), 67.
  * [36] D. T. Stoeva, P. Balazs, Invertibility of multipliers, Appl. Comput. Harmon. Anal., 33(2), (2012), 292–299.
  * [37] D. T. Stoeva, P. Balazs, Detailed characterization of conditions for the unconditional convergence and invertibility of multipliers, Sampl. Theory Signal Image Process., 12(2-3), (2013), 87–125.
  * [38] D. T. Stoeva, P. Balazs.  Canonical forms of unconditionally convergent multipliers.  J. Math. Anal. Appl., 399, (2013), 252–259.
  * [39] D. T. Stoeva, P. Balazs, Riesz bases multipliers, In M. Cepedello Boiso, H. Hedenmalm, M. A. Kaashoek, A. Montes-Rodríguez, and S. Treil, editors, Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation, vol 236, Oper. Theory Adv. Appl., 475-482. Birkhäuser, Springer Basel, 2014.
  * [40] D. T. Stoeva, P. Balazs, On the dual frame induced by an invertible frame multiplier, Sampl. Theory in Signal and Image Process., 15, (2016), 119–130.
  * [41] D. T. Stoeva, P. Balazs, Commutative properties of invertible multipliers in relation to representation of their inverses, In Sampling Theory and Applications (SampTA), 2017 International Conference on, 288–293. IEEE, 2017.
  * [42] D. T. Stoeva, P. Balazs, A survey on the unconditional convergence and the invertibility of multipliers with implementation, In: Sampling - Theory and Applications (A Centennial Celebration of Claude Shannon), S. D. Casey, K. Okoudjou, M. Robinson, B. Sadler (Ed.), Appl. Numer. Harmon. Anal. Series, Springer, 2020.