# BVI-Lowlight: Fully Registered Benchmark Dataset for Low-Light Video
Enhancement

Nantheera Anantrasirichai, Ruirui Lin, Alexandra Malyugina, and David
Bull111This work was supported by the UKRI MyWorld Strength in Places
Programme (SIPF00006/1) and Bristol+Bath Creative R+D under AHRC grant
(AH/S002936/1)
Visual Information Laboratory
University of Bristol

###### Abstract

Low-light videos often exhibit spatiotemporal incoherent noise, leading to
poor visibility and compromised performance across various computer vision
applications. One significant challenge in enhancing such content using modern
technologies is the scarcity of training data. This paper introduces a novel
low-light video dataset, consisting of 40 scenes captured in various motion
scenarios under two distinct low-lighting conditions, incorporating genuine
noise and temporal artifacts. We provide fully registered ground truth data
captured in normal light using a programmable motorized dolly, and
subsequently, refine them via image-based post-processing to ensure the pixel-
wise alignment of frames in different light levels. This paper also presents
an exhaustive analysis of the low-light dataset, and demonstrates the
extensive and representative nature of our dataset in the context of
supervised learning. Our experimental results demonstrate the significance of
fully registered video pairs in the development of low-light video enhancement
methods and the need for comprehensive evaluation. Our dataset is available at
DOI:10.21227/mzny-8c77.

##  1 Introduction

Low-light video footage is problematic due to the interactions between the
aperture, shutter speed and ISO parameters which cause distortions,
particularly under extreme lighting conditions. This is due to the inverse
correlation of increased photon noise with decreasing light intensity where
increased sensor gain amplifies noise. Secondary characteristics such as white
balance and color effects will also be affected and will require correction.
While these distortions cause a reduction in perceived visual quality, the
associated artefacts also pose significant challenges to machine learning
tasks such as classification and detection. This is especially the case when
one considers the effect of adversarial examples on some deep learning
networks.

Figure 1: Scene examples with varying light levels (10% and 20% low-light,
100% normal light) and different motion profiles (bottom row). From left to
right columns: Mario3 with a static background, Book1, and Mario1. The length
of the red lines across the normal-light frames represents the height of the
x-t planes in the bottom row.

The performance of low-light image enhancement algorithms has been
significantly improved in recent year, primarily through the application of
deep learning methods. Such techniques, however, cannot be applied directly to
low-light video content as their application on a frame-by-frame basis causes
temporal inconsistencies. Optimization of these methods for motion video is,
in turn, limited by the need for large and diverse datasets with aligned
ground truth information. However, low-light video enhancement is an ill-posed
problem, meaning that clean ground truth information with accurate brightness
is not available.

Improving video footage shot under low-light conditions is highly required and
is emerging as an important research field, particularly within the context of
deep learning methods. In general these solutions, especially supervised
learning architectures, require aligned and accurate ground-truth data to
support robust training. This however is limited for low-light video research.
To address this, researchers frequently resort to the use of synthesized
noise. Self-supervised learning methods, which do not rely on ground truth
data, can be applied to denoising tasks [1]. This is possible because the
details are randomly collapsed, and content priors are inhibited in the
videos. However, while these methods can achieve contrast enhancement to a
certain extent, akin to applying histogram stretching, they are ill-suited for
color correction tasks. The reason is that low-light images and videos lack
accurate color information, rendering color correction impossible. This has
been confirmed by empirical tests reported in [2]. Alternatively, a pseudo
reference with a unpaired learning strategy can be used [3], but the methods
generally are often complex and require large memory.

This paper presents a dataset of fully registered sequences between low-light
videos and their ground truth, shown in Fig. 1. It was captured over three
varying lighting conditions, including the 100% lighting being classed as
ground-truth. The dynamic scenes were created with programmable motorized
dolly, allowing repeatable motion profiles. Our dataset comprises a diverse
range of forty scenes, containing numerous contents, varying structures and
textures. This results in more than 30k ground-truth paired frames. It is
worth noting that normal light levels can be conveniently adjusted to achieve
the desired exposure, thereby creating pseudo ground-truth data to evoke
different cinematic moods and tones. Additionally, we chose to capture the
videos in the standard RGB format, whereas many datasets are in the raw Bayer
format. This decision makes our dataset more accessible to the research
community since consumer cameras typically do not provide raw videos.

We also present an exhaustive analysis of the low-light dataset, focusing on
noise, color correction, contrast enhancement in both spatial and temporal
information. Moreover, we demonstrate the extensive and representative nature
of our dataset in the context of supervised learning by employing it to train
state-of-the-art models for image and video enhancement. Our experimental
results show that the significance of fully registered video pairs in the
development of low-light video enhancement methods and comprehensive
evaluation.

In conclusion, our main contributions are as follows:

  * •

We present the high-quality video dataset captured in motion under three
distinct real lighting conditions, offering 40 different dynamic scenes with
diverse content.

  * •

Our dataset is the fully registered videos (spatially aligned frames between
different lighting videos) in each scene, facilitating supervised learning and
enabling full-reference quality assessment.

  * •

Our dataset provides two specific light levels, rather than random variations
in brightness. This design allows for a more in-depth analysis of the effects
of different light levels, contributing to a better understanding of their
characteristics and influence on deep learning architectures.

  * •

We conducted extensive experiments to demonstrate the impact of fully-
registered videos, diverse content, and motion on the performance of low-light
video enhancement methods.

##  2 Existing low-light video datasets

This section discusses datasets available for low-light video enhancement.
While the number of datasets captured under real low-light conditions is
limited, synthetic samples have expanded research activities in this field.
However, it is important to note that synthetic distortions are often
oversimplified, as mentioned earlier, low-light conditions combine multiple
distortion types.

###  2.1 Real datasets

Several ‘image’-based datasets exist that can be used to support denoising and
other inverse problems. These include the UCID-v2 corpus [4], the Darmstadt
Noise Dataset [5], the Smartphone Image Denoising Database (SIDD) [6]. There
are a couple of low-light video datasets captured in static scenes, allowing
for the creation of ground truth. This includes BVI-LOWLIGHT, which offers
true ISO noise in videos captured at 30 frames per scene [7], and DRV (also
referred to as SMID) [8], capturing 60-100 frames per scene. The DRV also
provides dynamic videos without ground truth, similar to the BBD-100K dataset
[9]. Notably, only three datasets that feature motion and provide ground truth
to date:

i) SMOID [10]: The primary purpose of this dataset is for video surveillance.
The dataset consists of 179 pairs of street view videos containing moving
vehicles and pedestrians. These video pairs were captured simultaneously using
a beam splitter, and an ND (Neutral-density) filter was inserted to limit the
amount of light in one camera to simulate low-light environment. Only the
overlapping area was used, resulting in a frame resolution of 1800$\times$1000
pixels. Each video has 200 frames with a GBRB Bayer pattern. Unfortunately,
this dataset is not publicly available.

ii) RViDeNet [11]: It has been introduced specifically for video denoising
purposes, where the videos were captured at 5 different ISO levels, ranging
from 1600 to 25600. Each video contains only 7 frames with a resolution of
1920$\times$1080 pixels in Bayer format. They were captured using the stop
motion technique, so there is no motion blur as in reality. Also, the
displacement between frames is very large resulting unnatural footage. The
corresponding clean frame is obtained by averaging multiple noisy frames,
resulting in a clean frame that is still dark. This means that this dataset
cannot be used for brightness enhancement and color correction, as required
for low-light video enhancement.

iii) SDSD [12]: The video pairs were captured using a motorised dolly.
However, their video frames are not correctly aligned, making them unsuitable
for supervised learning or objective evaluation. Our testing experiments have
confirmed this issue. The dataset comprises 150 video pairs featuring both
indoor and outdoor scenes, each consisting of 100-300 frames per video and
having a resolution of 1920$\times$1080 pixels. Although they claimed that
their darker videos were created with an ND filter and adjusted ISO settings,
we found inconsistencies in the lighting conditions, as some video pairs
exhibit different shadows. Unfortunately, there are no specific details
available regarding the camera settings they used. Moreover, several reference
videos are overexposed, making it impossible to recover the details in these
areas. This is confirmed by their analysis, which reveals that some low-light
patches have higher intensity than their corresponding normal-light patches.
Examples of the SDSD datasets are shown in Fig. 2.

iii) DID [13]: There are 413 pairs of videos, each with a resolution of
2560$\times$1440, captured with five different cameras. The dynamic scenes
were created using an electric gimbal that rotated around the camera axis.
This setup however results in only one type of motion–panning. The video
brightness was adjusted using an ND filter.

Figure 2: Examples of faulty SDSD video pairs, where the ground truth and the
low-light frames are on the left and the right, respectively. (Top) The
overexposed areas in the ground truth frame. (Bottom) The unaligned pair.

Table 1 shows the existing low-light datasets. These current datasets are
obviously limited to develop the reliable tools for video enhancement, either
because they are too small or because they lack realistic distortions or
ground truths for moving objects. This consequently impacts their suitability
for training and validating new networks. Moreover, the low-light condition of
most datasets providing ground truth were created with ND filters. While ND
filters are intended to reduce the amount of light entering your camera
uniformly across all levels of intensity in the scene, they may exhibit
variations in spectral transmittance, potentially leading to color shifts.
Moreover, ND filters can introduce optical aberrations, reduced sharpness,
reflections and lense flare, and exhibit potential degradation over time,
impacting image quality and reliability in the experiments [14].

Table 1: Low-light video dataset comparison. S and D indicate static and dynamic scenes, respectively. 1D means one type of movement. D∗ means stop motion. †SMOID dataset is not publicly available.  Name | Light | Motion | Frames | GT | Register
---|---|---|---|---|---
BVI-LOW [7] | Real | S | 13,200 | ✓ | ✓
DRV [8] | Real | S | 20,847 | ✓ | ✓
DRV [8] | Real | D | 2,397 | ✗ | ✗
BBD-100K [9] | Real | D | - | ✗ | ✗
SMOID† [10] | ND filter | 1D | 35,800 | ✓ | ✓
RViDeNet [11] | Real |  D∗ | 385 | ✓ | ✓
SDSD [12] | ND filter | 1D | 37,500 | ✓ | Mix
DID [13] | ND filter | 1D | 41,038 | ✓ | ✓
Ours | Real | D | 31,800 | ✓ | ✓

###  2.2 Synthetic datasets

The simplest approach applies Gaussian noise to the clean image as additive
noise [15]. Basic models like these do not effectively capture the noise found
in natural images, especially in low-light conditions, where photon noise is
dominant [7]. Authors in [16] have analyzed their low-light videos captured
with Canon 5D mark III and found that dynamic streak noise (DSN) is dominate.
They generate the synthetic noise based on DSN, clipping effect, shot noise
and photoelectrons. Note that we do not observe DSN in our videos captured
with a Sony Alpha 7SII camera or professional cameras, e.g. RED and ARRI used
by filmmakers. SIDGAN [17] has been proposed to map videos found ‘in the wild’
into a low-light domain, producing synthetic training data. This work is based
on dual CycleGAN, which map videos captured with short exposure (low light)
into the synthetic long exposure domain (normal light). The authors found that
training the model with a combination of real and synthetic data achieves the
best performance in video enhancement.

##  3 State-of-the-art methods for video enhancement

With recent advances in deep learning technologies, ‘image’ enhancement has
made significant progress. In contrast, learning-based video enhancement
techniques are still in their early stages, primarily due to more unknown
parameters, limitations in datasets, computational complexity, and high memory
requirements. Applying image-based techniques sequentially to each frame in
the video is a common approach but often results in temporal inconsistencies.
An in-depth survey of learning-based methods for low-light image and video
enhancement proposed between 2017 and 2021 can be found in [18], which also
shows that there were only a few methods specifically proposed for video
enhancement.

###  3.1 Low-light image enhancement

We have reviewed recent low-light image enhancement methods that could
potentially be extended to video sequences. In [19], the noise map is simply
estimated by subtracting the noisy image from the blur-filtered image, and it
then guides the attention mechanism in the transformer. In [20], a self-
calibrated illumination (SCI) framework has been proposed to learn the
residual representation between the illumination and low-light observation.
Recently, deep unfolding approaches have garnered significant interest. An
adaptive total variation regularization method has been proposed in [21],
employing a network to predict noise levels. The technique in [22] integrates
the Retinex theory model, where the image is an element-wise multiplication of
reflectance and illumination, with learnable physical priors for reflectance
and illumination.

###  3.2 Low-light video enhancement

Multiple input frames are typically used, denoted as $I_{t+i},i\in[-N,N]$,
where $N$ represents the number of neighboring frames at time $t$. In the case
of fully registered frames, such as those from static scenes, averaging pixel
values across the temporal dimension results in smoothness and noise
reduction. However, this simple strategy is challenged by moving objects or
camera motion. SMOID [10] modified UNet to handle multiple frames, assuming
that the multiple convolutional layers could handle the displacement between
frames due to moving objects. However, this method may not effectively handle
fast motion. Therefore, several methods include an alignment module, which
aligns the feature maps of neighboring frames with those of the current frame
[12, 23]. Earlier work exploits LSTM (Long Short-Term Memory) [16], whilst
later, Deformable Convolution Networks (DCN) [24] are more commonly employed
for this purpose. The alignment module still sometimes struggles to perform
perfect motion compensation, leading to artifacts when combining features from
multiple frames. To address this issue, the authors in [23] include
iteratively re-weighting features to minimise merging errors. Authors in [25]
proposed synthetic events from the video to adaptively combine multiple frames
according to speeds of moving objects. Attention modules are utilised in
fusing synthetic guided event and spatial features.

Most methods extract features using ResNet, UNet or a combination of them. DRV
[8] has two ResUNet networks that are shared weights, which are trained on the
two random frames of static videos, based on an assumption that the model can
extend to dynamic videos. Similarly, SIDGAN [17] trains their networks with
shared weights. SDSD [12] separates noise $n_{t}$ and illumination $S_{t}$,
generating output $\hat{I}_{t}=(I_{t}-n_{t})/S_{t}$. Some methods require
Bayer raw video inputs [8, 10], which are not suitable for many consumer
cameras.

As availability of paired datasets is limited, there are also some attempts
that use unpaired training strategy, e.g. using CycleGAN [3]. This
architecture is also employed to map the low-light RAW to RGB format [17].

###  3.3 Video restoration

Deep learning-based techniques for video restoration have demonstrated their
effectiveness in tasks like denoising, deblurring, and dehazing. Typically,
these networks are trained to address the specific type of distortions.
However, challenges arise when natural videos, particularly those captured in
challenging environments like low light, exhibit a combination of various
distortion types. One of the reasons might be due to the lack of reliable
datasets. Therefore, we believe that it is valuable to mention and assess the
performance of these networks on our datasets. One notable state-of-the-art
model is EDVR [26], which employs DCN and attention-based modules to integrate
spatial and temporal information. Their applications are super-resolution and
deblurring. Vision transformer for videos [27] employs temporal mutual self-
attention, which is focused on mutual alignment between two neighbouring
frames, while self attention is used for feature extraction. Subsequently, the
same researchers integrated recurrent feature refinement into the framework,
allowing the target frame to benefit from the restoration of previous frames
and reducing memory requirements [28]. The networks have been tested on low-
resolution, blurred, and noisy videos separately.

##  4 Proposed low-light dataset

###  4.1 Environment settings

Acquisition of the new dataset was conducted in a dedicated studio, designed
to permit full and precise control of the environment’s light intensities.
Four Cineo MavX lights, with up to 8,000 lumens output, were configured in
Digital Multiplex (DMX) mode and wired to a Zero 88 FLX S24 lighting console.
The color temperature (CCT) was set to 6,500 Kelvin and the range of lighting
intensities. Each scene contains a ‘Datacolor SpyderCheckr24: 24 color Patch
and Grey Card’, typically used for camera calibration and as a white balance
target. This provides a set of homogeneous patches as a reference.

We used a Sony Alpha 7SII Wi-Fi enabled camera, capable of recording 4K
resolution (3840$\times$2160 pixels) video footage. The repeat movement was
controlled using a Kessler CineDrive shuttle dolly system. The CineDrive
system, mounted on the dolly allowed additional horizontal camera movement and
rotation in three axes, as shown in 3. The frame rate was set to 25 fps, which
led to a fixed shutter speed of 1/50, as recommended by professionals to
achieve a shutter angle of 180 degrees (equivalent to half the frame rate).
The aperture was set at F6.8, which fell within the middle range of the
camera’s optimal aperture setting, producing the highest image quality. In the
low-light environments, the ISO was fixed at 800, the optimal value
recommended by the manufacturer. It is worth noting that with the Sony camera
we used, in-camera noise removal occurs when a higher ISO is employed.

Figure 3: Setting environment showing Sony Alpha 7S II camera in ‘angle’
position, mounted on CineDrive system, with scene in the background.

Camera triggering to record a scene is performed using Sony’s Imaging Edge
mobile phone app, where the phone is connected to the camera via Wi-Fi. This
app also supports the configuration of settings such as aperture, white
balance and shutter speed. The use of this remote app avoids unwanted shadows
in the footage and sensor misalignments that could arise if the camera was
manually triggered.

In this dataset of video pairs, we recorded low-light videos at 10% and 20% of
normal lighting levels (100%), as indicated by the Zero 88 FLX S24 light
controller. These videos are provided in full HD resolution (1920x1080
pixels), with cropping applied to focus on the region of interest. To ensure
that the objects were visible throughout the entire length of the clipped
videos, we manually selected the start and stop points.

###  4.2 Ground truth generation

The primary sources of misalignment are related to the mechanical tolerances
of the shuttle dolly and the acceleration of the control motor. Despite
careful setup of the mechanical system, there is a possibility of frame
misalignment between any two identical motion profiles. We keep the low-light
videos as they are because adjusting them could alter the noise
characteristics and motion blur. Therefore, we generate pseudo ground truth
that is pixel-wise registered to each frame of the low-light videos as
follows.

The ground truth videos were recorded under normal lighting condition. To
minimize misalignment between the ground truth and the low-light videos, we
repeated the dolly movement to create three videos for slow motion and five
videos for fast motion. This introduced slight pixel shifts among them,
ensuring that the best match to the low-light video had the smallest possible
misalignment. In most cases, the best matching video remained consistent
throughout the entire video. However, in a few video pairs, the best matching
video changed due to a change in the dolly’s direction in the middle of the
video, such as moving from left to right and then turning back to the left.

The best video matching and frame synchronization were performed by firstly
cropping the video by 100 pixels from the borders of the frame to avoid the
influence of the background. We subsequently enhanced the frame $t$ of the
low-light video $J$ by matching its histogram of each color channel to the
histogram derived from the normal-light frame $I^{m}_{n}$ of the video $m$ at
frame $n$, i.e. $\hat{J}_{t}=\mathcal{H}(J_{t},I^{m}_{n})$, where
$\mathcal{H}$ is a histogram matching function with 255 bins for images with
an 8-bit depth. This can be seen as contrast enhancement and color correction
with respect to the information from the reference (normal light in our case).
Then, the best-aligned reference frame (normal light level) was found by
minimising the mean absolute error between $\hat{J}_{t}$ and $I^{m}_{n}$,
where $n\in[t-N_{w},t+N_{w}]$, $N_{w}$ is a predefined searching window range,
$m\in[1,N_{v}]$, and $N_{v}$ is the total number of videos repeatedly captured
at the same normal-light scene. Fig. [fig:bunny] shows error maps depicting
the differences between the low-light frame and the normal-light frame before
and after frame synchronization.

Figure 4: Moving bunny scene with static background showing pixel value
difference between the normal-light frame and the adjusted low-light frame
before and after frame matching. Gray indicates zero error.

###  4.3 The dataset

In total, we acquired 40 scenes of two different low light levels, including 6
scenes of static background. This results in 31,800 image pairs with various
motion settings, as detailed in Table 6. Example scenes are presented in Fig.
5, where 100% indicates normal light which can be used as reference or ground
truth. The number of frames varied based on the dolly’s speed.

We captured both static backgrounds with moving objects and complete dynamic
scenes. For the static background scenes, we placed the objects on the dolly,
and the camera on the tripod. In contrast, for creating more dynamic content,
we mounted the camera on the dolly, creating in both linear and nonlinear
motions. Three different camera positions and movements relative to the
dolly’s direction were configured, denoted as follows in Table 6: i) parallel,
where the camera axis remained perpendicular to the dolly’s movement
direction, ii) angle, where the camera axis was positioned at a 30-degree
angle to the dolly’s movement direction (as shown in Fig. 3), and iii)
rotation, indicating camera rotation in its yaw angle while the dolly was in
motion. Dolly speeds were set to be constant or varied after clipping, denoted
as linear and nonlinear. Fig. 1 displays various motions captured in the
videos. The x-t planes of three example videos reveal different movements.
Mario3 exhibits linear motion of the foreground object with a static
background. In the dynamic scene of Book1, there is a transition from fast
motion at the beginning, slowing down in the middle, and speeding up towards
the end of the video. In the Mario1 scene, there is nonlinear movement from
left to right and then a return to the left of the scene.

Fig. 6 reveals the noise present in the low-light scenes (for visualization,
we applied histogram matching). It’s worth noting that color correction is not
needed extensively, as we have included the standard color palette in the
scene, which should also assist in the training process. The bottom row shows
a magnified, cropped image of the mug in the Kitchen scene. The letters (‘A
cup of’) in the low-light video appear thicker compared to those in the
normal-light one, which is a result of motion blur.

Figure 5: The proposed dataset: (From left to right) flowers, soft toy,
wrapping, wood board, hats. (From top to bottom) Light levels of 10%, 20%, and
normal light (100%). Figure 6: Cropped images (lego and kitchen scenes at
350$\times$350 and 140$\times$140 pixels, respectively) with histogram
matching to the reference (normal light) to visualise noise at different
levels of light. (From left to right) Light levels of 10%, 20%, and normal
light (100%).

##  5 Experiments

This section examines the importance of fully registered video pairs in the
development of low-light video enhancement methods. We analyze distortions in
low-light video sequences and compare our proposed datasets with existing
ones. We achieve this by using them to train several state-of-the-art methods
for low-light enhancement and video restoration. In all experiments that used
our dataset, we randomly selected 32 scenes for training and allocated the
remaining scenes for testing. These training and testing sets were fixed to
all experiments.

###  5.1 Importance of fully registered data

The training process of supervised learning optimizes the model by minimizing
loss. The most commonly used loss functions in image and video enhancement are
$\ell_{1}$, $\ell_{2}$, charbonnier, SSIM, and VGG loss [29]. While $\ell_{1}$
and $\ell_{2}$ require pixel-wise alignment between the predicted output and
the ground truth, SSIM and VGG losses allow for a slight shift between them.

We demonstrate the importance of a fully registered dataset for supervised
learning approaches in image enhancement by retraining three network
architectures: UNet [30], Conditional GAN [31], and Transformer (SwinIR [32])
with our dataset before and after frame synchronization, as explained in
Section 4.2. Prior to synchronization, the maximum misalignment is 20 pixels
(considering that the frame resolution is 1,920$\times$1,080 pixels). We also
conducted experiments with synthetic misalignments of 3, 5, and 10 pixels
between the input and ground truth and the results are reported in
supplementary material.

Table 2 shows the performance impact of low-light enhancement when the models
are trained with either non-registered or registered datasets. As expected,
performance decreases when trained with unregistered datasets, with a
noticeable effect when using the $\ell_{1}$ loss. Fig. 7 shows the results of
testing dynamic scenes in our dataset and the DRV dataset, both trained with
our dataset before and after alignment (the top row used SSIM loss, while the
bottom row used $\ell_{1}$ loss). The results clearly exhibit artifacts, which
occur because the network cannot align the predicted output with the ground
truth to minimize the loss. The DRV dataset does not provide ground truth but
offers a long-exposure version for brightness reference.

Table 2: Comparison of results (PSNR/SSIM) using non-registered (No) and registered (Yes) datasets with different loss functions for training. loss | $\ell_{1}$ | SSIM | VGG
---|---|---|---
Register? | No | Yes | No | Yes | No | Yes
PSNR
UNet | 20.05 | 20.72 | 19.45 | 20.37 | 19.89 | 20.94
CGAN | 20.75 | 26.60 | 23.91 | 26.14 | 24.60 | 28.41
SwinIR | 20.69 | 23.62 | 18.45 | 22.04 | - | -
SSIM
UNet | 0.760 | 0.761 | 0.731 | 0.785 | 0.752 | 0.753
CGAN | 0.702 | 0.862 | 0.786 | 0.861 | 0.796 | 0.911
SwinIR | 0.693 | 0.745 | 0.677 | 0.732 | - | -
Figure 7: Comparison of training CGAN [31] with the unregistered and
registered datasets. Top and bottom rows are the cropped testing dynamic
scenes of our dataset and DRV dataset, respectively.

###  5.2 Influence of different light levels in datasets

Our dataset provides two light levels in low-light environments. This allows
us to investigate whether the improved video quality depends on the light
levels present in the training data. We trained our PCDUNet using data from
only the 10% lighting, 20% lighting, and both light levels. Then we tested
them with the same and cross light levels. The results are shown in 3. It is
obvious that the model trained with the same lighting level as the testing
data performs better than one trained with different lighting levels. However,
if the light level is unknown, the model trained with various light levels
exhibits more robustness, as shown in the third column of the table.

Table 3: Performance of low-light video enhancement method, PCDUNet, when training with different light levels test | 10% | 20% | 10%+20%
---|---|---|---
train | PSNR/SSIM | PSNR/SSIM | PSNR/SSIM
10% | 29.52/0.894 | 12.76/0.669 | 21.33/0.784
20% | 13.49/0.620 | 31.36/0.936 | 22.24/0.775
10%+20% | 17.81/0.728 | 31.17/0.932 | 24.52/0.845

###  5.3 Dataset comparison

We evaluated the training effectiveness of our datasets with low-light video
enhancement methods, including SMOID [10], SMID [8], SDSD [12], as well as
state-of-the-art video restoration techniques, such as EDVR [26], VRT [27],
and our PCDUNet, as utilized in a previous study. We compared the performance
of these models trained with our datasets to those trained with existing
datasets, namely DRV [8] and SDSD [12]. The RGB data of DRV222the data
available online are in standard RGB format with 8-bit depth appears
excessively green, which is not natural (see Fig. 8). Upon observation, we
noticed that the values of the green channel are approximately twice as high
as those in the other color channels. Consequently, we adjusted the green
channel by reducing its values by half to ensure a fair comparison when using
the networks that accept standard RBG inputs.

When dividing the data for training and evaluation, we followed to the
parameters outlined in the original papers. The training-to-testing ratios
were 73:27 for DRV and 84:16 for SDSD. In our dataset, we chose an
intermediary approach by randomly splitting our datasets into 80% for training
and 20% for testing. The training parameters were also aligned with those
outlined in the original papers, ensuring consistency across all datasets for
a fair comparison. The results, presented in Table 4, are the average
performance across three datasets.

Figure 8: DRV dataset [8] and our green channel adjustment Table 4: Performance comparison of the models trained with different datasets. The objective results shown here are the average of three testing datasets. Dataset |  DRV [8] |  SDSD [12] | Our dataset
---|---|---|---
Methods | PSNR/SSIM | PSNR/SSIM | PSNR/SSIM
Without postprocessing
UNet [30] | 13.73/0.328 | 15.89/0.617 | 19.44/0.710
CGAN [31] | 17.53/0.490 | 15.52/0.461 | 19.80/0.621
RIDNet [33] | 15.30/0.544 | 15.02/0.498 | 20.48/0.709
SwinIR [32] | 17.07/0.696 | 12.11/0.484 | 20.06/0.726
SDSD [12] | 15.29/0.542 | 15.01/0.496 | 16.03/0.616
PCDUNet | 16.67/0.512 | 16.11/0.636 | 20.80/0.757
Postprocessing with histogram matching to a reference
CGAN [31] | 29.67/0.836 | 28.26/0.800 | 32.91/0.890
SwinIR [32] | 28.31/0.839 | 26.68/0.789 | 30.86/0.908

###  5.4 Dataset comparison when lighting reference is available

As brightness perception in videos can be subjective, we conducted evaluations
considering a preferable brightness level as postprocessing. To do this, we
used the first frame from the ground truth sequence as the reference. We
performed histogram matching between this reference frame and the first frame
of the enhanced output. Subsequently, each subsequent enhanced frame was
adjusted concerning its previous frame, ensuring consistency throughout. Table
5 shows the results comparing postprocessing against scenarios with no
brightness reference. Interestingly, the PSNR and SSIM of the testing SDSD
dataset exhibit the most significant improvement across all models trained
with various datasets, with more than threefold increase. This suggests that,
when disregarding brightness considerations, enhancing the SDSD dataset is
notably more straightforward. Primarily, this trend arises due to the SDSD
dataset containing a smaller amount of noise compared to other datasets.
Moreover, its ground truth brightness significantly differs from the normal
light defined in other datasets, resulting the models perform worse when
trained with the SDSD dataset. Comparing training and testing on the same
dataset, the postprocessing improves the performances by approximately 25%,
200%, and 5% (average of PSNR and SSIM) when using DRV, SDSD and our dataset,
respectively. This implies that our dataset has better consistent brightness
of the ground truth.

Table 5: Performance comparison of CGAN [31] when trained on different datasets, reporting on individual testing datasets with and without postprocessing. | Test on
---|---
Train |  DRV [8] |  SDSD [12] | Our dataset
Without postprocessing
DRV [8] | 21.25/0.657 | 11.86/0.254 | 19.48/0.559
SDSD [12] | 17.76/0.518 | 9.56/0.307 | 19.26/0.560
Our dataset | 18.54/0.611 | 12.45/0.340 | 28.41/0.911
Postprocessing with histogram matching to a reference
DRV [8] | 27.33/0.810 | 36.01/0.904 | 25.67/0.794
SDSD [12] | 23.57/0.715 | 37.32/0.912 | 23.89/0.765
Our dataset | 26.67/0.812 | 40.32/0.933 | 31.78/0.925

##  6 Conclusion

This paper introduces a novel dataset for low-light video enhancement.
Comprising 40 dynamic scenes captured under two low-light levels, our dataset
ensures full registration in both spatial and temporal dimensions with the
corresponding normal-light videos which can be used as ground truth. This
benefits the development of supervised learning models and supports full-
reference quality assessment. We illustrate the effectiveness of our dataset
through a comprehensive study and experimental comparisons with existing low-
light video datasets.

Table 6: Scene descriptions: Frame denotes the total number of frames. Speed represents the approximate dolly speed. Camera describes the camera’s movement relative to the dolly, with ‘parallel’ and ‘angle’ denoting fixed angles, and ‘rotation’ indicating camera rotation in its yaw angle. Motion uses ‘linear’ and ‘nonlinear’ to distinguish constant and varied speeds, respectively. ‘(S)’ denotes static background. # | Name | Frames | Speed | Camera | Motion
---|---|---|---|---|---
1. | color chart | 270 | slow | parallel | linear
2. | Animals1 | 425 | slow | angle | linear
3. | Animals2 | 600 | slow | rotation | nonlinear
4. | color sticks | 350 | slow | parallel | linear
5. | Bunnies | 300 | fast | parallel | nonlinear
6. | Lego | 214 | fast | parallel | linear
7. | Hats | 650 | slow | parallel | nonlinear
8. | Soft toys | 400 | slow | parallel | nonlinear
9. | Kitchen | 257 | fast | rotation | nonlinear
10. | Messy toys | 210 | fast | parallel | nonlinear
11. | Gift wrap | 260 | fast | parallel | nonlinear
12. | Woods | 240 | fast | angle | linear
13. | Wires | 240 | fast | angle | linear
14. | color balls | 240 | fast | parallel | linear
15. | Flowers | 330 | mix | rotation | nonlinear
16. | Faces1 | 730 | slow | parallel | nonlinear
17. | Faces2 | 400 | fast | parallel | linear
18. | Books1 | 500 | fast | parallel | linear
19. | Books2 | 554 | fast | parallel | nonlinear
20. | Books3 | 380 | fast | parallel | linear
21. | Mario1 | 700 | fast | parallel | nonlinear
22. | Mario2 | 340 | fast | angle | linear
23. | Messy wires | 580 | slow | angle | nonlinear
24. | Figures1 | 450 | slow | parallel | nonlinear
25. | Figures2 | 200 | fast | rotation | nonlinear
26. | Cameras | 300 | fast | rotation | linear
27. | Cloths1 | 430 | fast | rotation | nonlinear
28. | Cloths2 | 330 | fast | mix | nonlinear
29. | Mix1 | 600 | mix | parallel | nonlinear
30. | Mix2 | 550 | slow | angle | linear
31. | Mix3 | 300 | slow | parallel | linear
32. | Mix4 | 450 | mix | parallel | nonlinear
33. | Mix5 | 200 | fast | rotation | linear
34. | Mix6 | 250 | fast | rotation | linear
35. | Bunny (S) | 480 | slow | parallel | linear
36. | Eeyore (S) | 480 | slow | parallel | linear
37. | Mario3 (S) | 370 | slow | parallel | nonlinear
38. | Mario4 (S) | 280 | fast | parallel | nonlinear
39. | Books4 (S) | 400 | slow | parallel | nonlinear
40. | Books5 (S) | 660 | mix | parallel | nonlinear

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# Point spectrum and hypercyclicity problem
for a class of truncated Toeplitz operators

Anton Baranov, Andrei Lishanskii  Anton Baranov
Department of Mathematics and Mechanics, St. Petersburg State University, 28,
Universitetskii prosp., St. Petersburg, 198504, Russia
<EMAIL_ADDRESS>Andrei Lishanskii
Department of Mathematics and Computer Science, St. Petersburg State
University, 29B, 14 line of the Vasilievsky Island, St. Petersburg, 199178,
Russia<EMAIL_ADDRESS>
###### Abstract.

In this note we discuss an open problem whether a truncated Toeplitz operator
on a model space can be hypercyclic. We compute point spectrum and
eigenfunctions for a class of truncated Toeplitz operators with polynomial
analytic and antianalytic parts. We show that, for a class of model spaces,
truncated Toeplitz operators with symbols of the form $\Phi(z)=a\bar{z}+b+cz$,
$|a|\neq|c|$, are not hypercyclic.

###### Key words and phrases:

hypercyclic operator, Toeplitz operator, model space, truncated Toeplitz
operator

###### 1991 Mathematics Subject Classification:

47A16, 47B35, 30H10

The results were obtained with the support of Ministry of Science and Higher
Education of the Russian Federation, agreement No 075-15-2019-1619.

##  1\. Introduction

A continuous linear operator $T$ on a separable Banach (or Frechét) space $X$
is said to be hypercyclic if there exists $x\in X$ such that the set
$\\{T^{n}x:\,n\in\mathbb{N}_{0}\\}$ is dense in $X$ (here
$\mathbb{N}_{0}=\\{0,1,2,\dots\\}$).

The first example of a hypercyclic operator in a Banach space setting was
given by S. Rolewicz [11] who showed that the operator $\alpha B$, where $B$
is the backward shift and $\alpha$ is an arbitrary complex number with
$|\alpha|>1$, is hypercyclic on $\ell^{p}$, $1\leq p<\infty$. It was shown
that many important classes of operators have this property. Among basic
examples of hypercyclic operators were Toeplitz operators with antianalytic
symbols.

###  1.1. Hypercyclic Toeplitz operators

As usual, $\mathbb{D}$ and $\mathbb{T}$ will denote the unit disk and the unit
circle, respectively, and $H^{2}$ will stand for the Hardy space in
$\mathbb{D}$. Recall that for a function $\psi\in L^{\infty}(\mathbb{T})$ the
Toeplitz operator $T_{\psi}:H^{2}\to H^{2}$ with the symbol $\psi$ is defined
as $T_{\psi}f=P_{+}(\psi f)$, where $P_{+}$ is the orthogonal projection from
$L^{2}(\mathbb{T})$ onto $H^{2}$. The following theorem, due to G. Godefroy
and J. Shapiro [8], describes all Toeplitz operators with antianalytic symbols
(i.e., $T_{\overline{\varphi}}$ where $\varphi\in H^{\infty}$).

######  Theorem.

(G. Godefroy, J. Shapiro, 1991) The Toeplitz operator
$T_{\overline{\varphi}}$, where $\varphi\in H^{\infty}$, is hypercyclic if and
only if $\varphi(\mathbb{D})\cap\mathbb{T}\neq\emptyset$.

It is well known that every Cauchy kernel
$k_{\lambda}(z)=(1-\bar{\lambda}z)^{-1}$, $|\lambda|<1$, is an eigenvector for
an antianalytic Toeplitz operator $T_{\overline{\varphi}}$ corresponding to
the eigenvalue $\overline{\varphi(\lambda)}$. Then sufficiency of the
condition $\varphi(\mathbb{D})\cap\mathbb{T}\neq\emptyset$ for hypercyclicity
follows immediately from the Godefroy–Shapiro Criterion (see [8], [4,
Corollary 1.10] or [9, Theorem 3.1]).

Analytic Toeplitz operators $T_{\varphi}$, $\varphi\in H^{\infty}$, are simply
multiplication operators. It is clear that they cannot be hypercyclic.

In [13] S. Shkarin posed the problem to describe hypercyclic Toeplitz
operators in terms of their symbols. This seems to be a difficult and, at the
moment, widely open problem. Shkarin gave a necessary and sufficient condition
for hypercyclicity in the case when the symbol is of the form
$\Phi(z)=a\overline{z}+b+cz$ (i.e., with tridiagonal matrix): $T_{\Phi}$ is
hypercyclic if and only if $|a|>|c|$ and

(1)
$(\mathbb{C}\setminus\Phi(\mathbb{D}))\cap\mathbb{D}\neq\emptyset,\qquad(\mathbb{C}\setminus\Phi(\mathbb{D}))\cap\widehat{\mathbb{D}}\neq\emptyset.$

Here $\widehat{\mathbb{D}}=\mathbb{C}\setminus\overline{\mathbb{D}}$.

Hypercyclicity of Toeplitz operators with the symbols of the form

$\Phi(z)=R(\overline{z})+\varphi(z),$

where $R$ is a polynomial or a rational function with poles outside
$\overline{\mathbb{D}}$ and $\varphi\in H^{\infty}$, was studied in [1, 3]. It
turned out that the valence of the meromorphic continuation of symbol $\Phi$
to $\mathbb{D}$, that is, $\Phi(z)=R\big{(}\frac{1}{z}\big{)}+\varphi(z)$ is
crucial for hypercyclicity. Also it was shown that the hypercyclicity problem
is closely related to cyclicity of certain families for some associated
analytic Toeplitz operator. A complete description of hypercyclic Toeplitz
operators in this class is not known, however, there is a number of conditions
sufficient for hypercyclicity. For the operators with symbols of the form
$\Phi(z)=a\overline{z}+\varphi(z)$ known necessary and sufficient conditions
almost meet:

######  Theorem.

([3]) Let $\Phi(z)=\frac{a}{z}+\varphi(z)$, where $\varphi\in H^{\infty}$,
satisfy (1).

1\. If $T_{\Phi}$ is hypercyclic, then $\Phi$ is univalent in $\mathbb{D}$;

2\. If $\varphi\in A(\mathbb{D})$ (disk-algebra) and $\Phi$ is injective in
$\overline{\mathbb{D}}$, then $T_{\Phi}$ is hypercyclic.

Thus, the difference is only in the injectivity on the boundary. Also it
should be noted that the second part of (1) (i.e.
$(\mathbb{C}\setminus\Phi(\mathbb{D}))\cap\widehat{\mathbb{D}}\neq\emptyset$)
is necessary for hypercyclicity of $T_{\Phi}$, while for the first part only a
weaker variant
$(\mathbb{C}\setminus\Phi(\mathbb{D}))\cap\mathbb{T}\neq\emptyset$ is known to
be necessary.

###  1.2. Truncated Toeplitz operators

Let $S^{*}f=T_{\bar{z}}f=\frac{f(z)-f(0)}{z}$ be the backward shift in
$H^{2}$. By the classical theorem of Arne Beurling, any $S^{*}$-invariant
subspace is of the form $K_{\theta}=H^{2}\ominus\theta H^{2}$ for some inner
function $\theta$ in $\mathbb{D}$. Subspaces $K_{\theta}$ are often referred
to as model spaces due to their role in Nagy–Foias model for contractions.

Truncated Toeplitz operators (TTO) are restrictions of usual Toeplitz
operators onto $K_{\theta}$. Namely, given a space $K_{\theta}$ and $\psi\in
L^{\infty}(\mathbb{T})$, define the operator $A_{\psi}:K_{\theta}\to
K_{\theta}$ by the formula

$A_{\psi}f=P_{\theta}(\psi f),\qquad f\in K_{\theta},$

where $P_{\theta}$ stands for the orthogonal projection from
$L^{2}(\mathbb{T})$ onto $K_{\theta}$. Formally, $A_{\psi}$ depends also on
$\theta$, but we always assume $\theta$ to be fixed and do not include it into
notations.

Even if some special cases of TTO-s were well studied for a long time (e.g.,
$A_{z}$ – the model operator of Nagy–Foias theory and its functions
$\varphi(A_{z})=A_{\varphi}$, $\varphi\in H^{\infty}$), a systematic study of
TTO-s was initiated by a seminal paper by D. Sarason [12]. Note that one can
consider operators with unbounded symbols (assuming there exists a bounded
extension of an operator defined on the dense subset of bounded functions in
$K_{\theta}$) and it may happen that a bounded TTO has no bounded symbol at
all (see, e.g., [2]. In what follows we consider the case of bounded symbols
only.

During the last decade the theory of truncated Toeplitz operators became an
active field of research (see the surveys [5, 7] and the references therein).
However, it seems that the spectral properties of TTO-s are far from being
well understood. E.g., to the best of our knowledge, it is not known whether
the point spectrum of a TTO can have nonempty interior.

A problem which apparently did not attract much attention yet is to understand
the dynamics of TTO-s and in particular their hypercyclicity. In fact, it is
not known whether hypercyclic TTO-s do exist.

######  Problem.

Do there exist hypercyclic truncated Toeplitz operators?

Note that, since $K_{\theta}$ is invariant with respect to antianalytic
Toeplitz operators, one has $A_{\bar{\varphi}}=T_{\bar{\varphi}}$ for
$\varphi\in H^{\infty}$. Also, recall that existence of an eigenvector for the
adjoint operator is a trivial obstacle for hypercyclicity. Combining this, one
comes to the following simple observation.

######  Observation.

If an inner function $\theta$ has a zero in $\mathbb{D}$, then for any
$\varphi\in H^{\infty}$ the truncated Toeplitz operators $A_{\varphi}$ and
$A_{\bar{\varphi}}$ are not hypercyclic.

###### Proof.

If $\theta(\lambda)=0$, then $k_{\lambda}(z)=(1-\bar{\lambda}z)^{-1}$ belongs
to $K_{\theta}$, and
$A_{\bar{\varphi}}k_{\lambda}=\overline{\varphi(\lambda)}k_{\lambda}$ whence
$A_{\varphi}=A_{\bar{\varphi}}^{*}$ is not hypercyclic. Also,

$A_{\varphi}\Big{(}\frac{\theta}{z-\lambda}\Big{)}=P_{\theta}\Big{(}\frac{\varphi\theta}{z-\lambda}\Big{)}=P_{\theta}\Big{(}\theta\frac{\varphi-\varphi(\lambda)}{z-\lambda}+\varphi(\lambda)\frac{\theta}{z-\lambda}\Big{)}=\varphi(\lambda)\frac{\theta}{z-\lambda}.$

Hence, $A_{\varphi}$ also has an eigenvector and so $A_{\bar{\varphi}}$ is not
hypercyclic. ∎

Thus, if we are interested in hypercyclicity of TTO-s with analytic or
antianalytic symbols, the only interesting case is the case of singular inner
function $\theta$. In this case the point spectrum is empty.

######  Problem.

Let $\theta$ be a singular inner function. Do there exist hypercyclic
truncated Toeplitz operators with analytic or antianalytic symbols?

One can start with the case of a sufficiently regular symbol $\varphi$, say in
the disk-algebra $A(\mathbb{D})$ or even analytic in a neighborhood of
$\overline{\mathbb{D}}$. In this case the spectrum of the operator
$A_{\varphi}$ is known to be $\varphi(\sigma(\theta))$ [10, Lecture III],
where $\sigma(\theta)$ stands for the boundary spectrum of $\theta$ (the set
of those $\zeta\in\mathbb{T}$ for which
$\liminf_{z\to\zeta,z\in\mathbb{D}}|\theta(z)|<1$). Thus, a necessary
condition for hypercyclicity is that
$\varphi(\sigma(\theta))\cap\mathbb{T}\neq\emptyset$.

######  Problem.

Let $\theta(z)=\exp\big{(}\frac{z+1}{z-1}\big{)}$ be the simplest atomic
singular inner function. Do there exist hypercyclic truncated Toeplitz
operators with a symbol $\varphi$ in $H^{\infty}$ or in $A(\mathbb{D})$? This
is equivalent to the following approximation problem: does there exist a
function $f\in K_{\theta}$ such that the set

$\overline{{\rm Span}}\,\\{\varphi^{n}f+\theta g:\ g\in
H^{2},n\in\mathbb{N}_{0}\\}=H^{2}\,?$

##  2\. Point spectrum of TTO-s with polynomial symbols

In this section we consider truncated Toeplitz operators with symbols of the
form

(2) $\Phi(z)=\sum_{k=1}^{N}a_{k}z^{-k}+\sum_{l=0}^{M}c_{l}z^{l},\qquad
M,N\in\mathbb{N},\ a_{N}\neq 0,$

and describe their point spectrum under some mild restrictions. It turns out
that the set of eigenvectors $\lambda$ has a rather curious structure
involving some polynomial dependence between the values of $\theta$ at the
roots of $\Phi-\lambda$.

######  Theorem 2.1.

Let $\Phi$ be given by (2) and let $z_{j}$, $j=1,\dots M+N$, be the zeros of
$\Phi-\lambda$. Assume that all $z_{j}$ are distinct and $|z_{j}|\neq 1$ for
any $j$. Then $\lambda$ is an eigenvalue of $A_{\Phi}$ if and only if there
exist polynomials $P_{1},P_{2}$ of degrees at most $M$ and $N$ respectively
such that the following conditions hold:

(3) $z_{j}^{N}P_{1}(z_{j})\theta(z_{j})+P_{2}(z_{j})=0,\qquad|z_{j}|<1,$

and

(4)
$z_{j}^{N}P_{1}(z_{j})+P_{2}(z_{j})\overline{\theta(1/\overline{z}_{j})}=0,\qquad|z_{j}|>1.$

In this case the corresponding eigenfunction is given by

$f(z)=\sum_{|z_{j}|<1}\beta_{j}z_{j}^{N}P_{1}(z_{j})\frac{\theta(z)-\theta(z_{j})}{z-z_{j}}+\sum_{|z_{j}|>1}\beta_{j}P_{2}(z_{j})\frac{1-\theta(z)\overline{\theta(1/\bar{z}_{j})}}{z-z_{j}}.$

for some complex coefficients $\beta_{j}$.

###### Proof.

We are looking for solutions of the equation $A_{\Phi}f=\lambda f$, $f\in
K_{\theta}$. Denote by $\varphi$ the analytic part of $\Phi$,
$\varphi(z)=\sum_{l=0}^{M}c_{l}z^{l}$. Note that
$A_{\Phi}f=P_{\theta}(T_{\Phi}f)$, and

$T_{\Phi}f=\varphi
f+\sum_{k=1}^{N}a_{k}\frac{f-\sum_{j=0}^{k-1}\frac{f^{(j)}(0)}{j!}z^{j}}{z^{k}}.$

Thus, $A_{\Phi}f=\lambda f$ is equivalent to the equation

$\varphi
f+\sum_{k=1}^{N}a_{k}\frac{f-\sum_{j=0}^{k-1}\frac{f^{(j)}(0)}{j!}z^{j}}{z^{k}}=\lambda
f+\theta h,$

where $h\in H^{2}$ or, equivalently,

$(\Phi-\lambda)f=\theta
h+\sum_{k=1}^{N}a_{k}\sum_{j=0}^{k-1}\frac{f^{(j)}(0)}{j!}z^{j-k}.$

Multiplying by $z^{N}$, we get

(5) $Qf=z^{N}\theta h+R,$

where $Q=z^{N}(\Phi-\lambda)$ is a polynomial of degree $M+N$ with the zeros
$z_{j}$ and
$R(z)=\sum_{k=1}^{N}a_{k}\sum_{j=0}^{k-1}\frac{f^{(j)}(0)}{j!}z^{N-k+j}$ is a
polynomial of degree at most $N-1$. Since $z_{j}$ are distinct we can write

(6) $\frac{1}{Q(z)}=\sum_{j=1}^{M+N}\frac{\beta_{j}}{z-z_{j}}$

for some coefficients $\beta_{j}$.

Assume that $\lambda$ is an eigenvector and so there exists a nontrivial $f\in
K_{\theta}$ satisfying (5). If $|z_{j}|<1$, then the right-hand side of (5)
must vanish at $z_{j}$ and we get $z_{j}^{N}\theta(z_{j})h(z_{j})+R(z_{j})=0$.
Hence, we have

(7) $\displaystyle f(z)$
$\displaystyle=\sum_{|z_{j}|<1}\beta_{j}\frac{z^{N}\theta(z)h(z)-z_{j}^{N}\theta(z_{j})h(z_{j})}{z-z_{j}}+\sum_{|z_{j}|<1}\beta_{j}\frac{R(z)-R(z_{j})}{z-z_{j}}$
$\displaystyle+\sum_{|z_{j}|>1}\beta_{j}\frac{z^{N}\theta(z)h(z)}{z-z_{j}}+\sum_{|z_{j}|>1}\beta_{j}\frac{R(z)}{z-z_{j}}.$

For $|z_{j}|<1$ we further have

$\frac{z^{N}\theta(z)h(z)-z_{j}^{N}\theta(z_{j})h(z_{j})}{z-z_{j}}=\theta(z)\frac{z^{N}h(z)-z_{j}^{N}h(z_{j})}{z-z_{j}}+z_{j}^{N}h(z_{j})\frac{\theta(z)-\theta(z_{j})}{z-z_{j}}.$

Note that the first term is in $\theta H^{2}$, while the second belongs to
$K_{\theta}$.

Since the degree of $R$ is at most $N-1$, we have the Lagrange interpolation
formula

(8)
$\frac{R(z)}{Q(z)}=\sum_{j=1}^{M+N}\beta_{j}\frac{R(z)}{z-z_{j}}=\sum_{j=1}^{M+N}\beta_{j}\frac{R(z_{j})}{z-z_{j}}.$

Finally, note that for $|z_{j}|>1$ we have

$\frac{1}{z-z_{j}}=\frac{\theta(z)\overline{\theta(1/\bar{z}_{j})}}{z-z_{j}}+\frac{1-\theta(z)\overline{\theta(1/\bar{z}_{j})}}{z-z_{j}},$

where, again, the first term is in $\theta H^{2}$, while the second is in
$K_{\theta}$. Combining all these observations we conclude that
$f=f_{1}+\theta f_{2}$ where $f_{1}\in K_{\theta}$ and $f_{2}\in H^{2}$ are
given by

(9) $\displaystyle f_{1}(z)$
$\displaystyle=\sum_{|z_{j}|<1}\beta_{j}z_{j}^{N}h(z_{j})\frac{\theta(z)-\theta(z_{j})}{z-z_{j}}+\sum_{|z_{j}|>1}\beta_{j}R(z_{j})\frac{1-\theta(z)\overline{\theta(1/\bar{z}_{j})}}{z-z_{j}},$
$\displaystyle f_{2}(z)$
$\displaystyle=\sum_{|z_{j}|<1}\beta_{j}\frac{z^{N}h(z)-z_{j}^{N}h(z_{j})}{z-z_{j}}+\sum_{|z_{j}|>1}\beta_{j}\frac{R(z_{j})\overline{\theta(1/\bar{z}_{j})}}{z-z_{j}}.$

However, by our assumption, $f\in K_{\theta}$, whence $f_{2}=0$. In view of
(6) this is equivalent to

(10)
$\frac{z^{N}h(z)}{Q(z)}=\sum_{|z_{j}|<1}\beta_{j}\frac{z_{j}^{N}h(z_{j})}{z-z_{j}}-\sum_{|z_{j}|>1}\beta_{j}\frac{R(z_{j})\overline{\theta(1/\bar{z}_{j})}}{z-z_{j}}.$

Hence $z^{N}h$ is a polynomial of degree at most $M+N-1$ and since $h$ is
analytic at $0$ we conclude that $h$ is a polynomial of degree at most $M-1$.
If we put $P_{1}=h$ and $P_{2}=R$, then the condition
$z_{j}^{N}\theta(z_{j})h(z_{j})+R(z_{j})=0$ for $|z_{j}|<1$ is equivalent to
(3). Comparing the residues at $z_{j}$ with $|z_{j}|>1$ in (10) we conclude
that $z_{j}^{N}h(z_{j})=-R(z_{j})\overline{\theta(1/\bar{z}_{j})}$ which is
equivalent to (4).

To prove that conditions (3) and (4) are sufficient for $\lambda$ to be an
eigenvector for $A_{\Phi}$, we reverse the arguments. Assume that there exists
$P_{1}$ and $P_{2}$ as in (3) and (4) and let $h=P_{1}$, $R=P_{2}$. Then the
interpolation formula (10) holds and so the function $f_{2}$ in (9) is zero.
Define $f_{1}$ by (9). Combining the formula for $f_{1}$ with (8) we see that
$f=f_{1}=f_{1}+\theta f_{2}$ satisfies the equality (7). Using the fact that,
$z_{j}^{N}\theta(z_{j})h(z_{j})+R(z_{j})=0$ for $|z_{j}|<1$ we, finally
conclude that $Qf=z^{N}h\theta+R$. Dividing by $z^{N}$ and comparing the
coefficients at negative powers it is easy to see that $R$ must coincide with
$\sum_{k=1}^{N}a_{k}\sum_{j=0}^{k-1}\frac{f^{(j)}(0)}{j!}z^{N-k+j}$ where
$a_{k}$ are the coefficients at $z^{-1}$ in (2). ∎

######  Remark.

Note that for some configurations of zeros with respect to the unit circle,
the conditions (3) and (4) may be never satisfied. In particular, as is shown
below, for a three-term TTO $\lambda$ is never an eigenvalue if $z_{1}$ and
$z_{2}$ lie in different components of $\mathbb{C}\setminus\mathbb{T}$.

######  Remark.

The above results remain true if there exist zeros of $\Phi-\lambda$ with
$|z_{j}|=1$, but $z_{j}\notin\sigma(\theta)$ and so $\theta$ is analytic in a
neighborhood of $z_{j}$. Such $z_{j}$ can be included in any of the conditions
(3) or (4) since in this case
$\theta(z_{j})=1/\overline{\theta(1/\overline{z}_{j})}$. The same is true if
$z_{j}\in\sigma(\theta)$ but $|\theta(z_{j})|=1$ and $\theta$ has a finite
angular derivative at $z_{j}$ ($z_{j}$ is a Julia–Carathéodory point for
$\theta$) and so the space $K_{\theta}$ contains the reproducing kernel at
$z_{j}$. It is not, however, clear to us whether $\lambda$ can be an
eigenvalue if one of $z_{j}$ lies on $\mathbb{T}$ and is not a
Julia–Carathéodory point for $\theta$.

##  3\. Three-term truncated Toeplitz operators

In this section we consider the case when $\Phi(z)=a\bar{z}+b+cz$. In this
case the formulation of Theorem 2.1 will be substantially simplified.

######  Corollary 3.1.

Let $\Phi(z)=a\bar{z}+b+cz$, $a,c\neq 0$, and assume that the zeros
$z_{1},z_{2}$ of the function $\Phi-\lambda$ are distinct and satisfy
$|z_{j}|\neq 1$, $j=1,2$. Then $\lambda$ is an eigenvalue for $A_{\Phi}$ if
and only if either

1\. $|z_{1}|,|z_{2}|<1$ and $z_{1}\theta(z_{1})=z_{2}\theta(z_{2})$,
or

2\. $|z_{1}|,|z_{2}|>1$ and
$\frac{1}{z_{1}}\theta\Big{(}\frac{1}{\bar{z}_{1}}\Big{)}=\frac{1}{z_{2}}\theta\Big{(}\frac{1}{\bar{z}_{2}}\Big{)}$.

###### Proof.

Since in this case $M=N=1$, the polynomials $P_{1}$ and $P_{2}$ in (3) and (4)
are just constants. If both $z_{1},z_{2}\in\mathbb{D}$, we have
$P_{1}z_{1}\theta(z_{1})+P_{2}=P_{1}z_{2}\theta(z_{2})+P_{2}=0$ which implies
$z_{1}\theta(z_{1})=z_{2}\theta(z_{2})$. Analogously, if
$z_{1},z_{2}\in\widehat{\mathbb{D}}$, then
$P_{1}z_{1}+P_{2}\overline{\theta(1/\overline{z}_{1})}=P_{1}z_{2}+P_{2}\overline{\theta(1/\overline{z}_{2})}=0$,
whence
$z_{1}^{-1}\overline{\theta(1/\overline{z}_{1})}=z_{2}^{-1}\overline{\theta(1/\overline{z}_{2})}$.

Finally, consider the case $|z_{1}|<1$, $|z_{2}|>1$. Then
$P_{1}z_{1}\theta(z_{1})+P_{2}=0=P_{1}z_{2}+P_{2}\overline{\theta(1/\overline{z}_{2})}$,
whence

$z_{1}\theta(z_{1})\overline{\theta(1/\overline{z}_{2})}=z_{2},$

an impossible equality since its left-hand side in $\mathbb{D}$ and
$|z_{2}|>1$.

Conversely, equalities $z_{1}\theta(z_{1})=z_{2}\theta(z_{2})$ and,
respectively,
$\frac{1}{z_{1}}\theta\Big{(}\frac{1}{\bar{z}_{1}}\Big{)}=\frac{1}{z_{2}}\theta\Big{(}\frac{1}{\bar{z}_{2}}\Big{)}$
imply the existence of the polynomials $P_{1}$ and $P_{2}$ of degree zero
(constants) such that (3) and(4) are satisfied and so $\lambda$ is an
eigenvector by Theorem 2.1. ∎

We excluded the case of multiple zeros of $\Phi-\lambda$ to avoid technical
difficulties. In this case our polynomial dependencies (3) and (4) will
involve also the values of the derivatives of $\theta$ at the multiple zeros.
However, in the case of symbols of the form $\Phi(z)=a\bar{z}+b+cz$ the answer
is easy.

######  Corollary 3.2.

Let $\Phi(z)=a\bar{z}+b+cz$, $a,c\neq 0$, and assume that $\Phi-\lambda$ has a
zero $z_{0}$ in $\mathbb{D}$ of multiplicity 2. Then $\lambda$ is an
eigenvalue for $A_{\varphi}$ if and only if
$z_{0}\theta^{\prime}(z_{0})+\theta(z_{0})=0$.

Note that if $z_{1},z_{2}$ are zeros of
$z(\Phi(z)-\lambda)=cz^{2}+(b-\lambda)z+a$, then $z_{1}z_{2}=a/c$. Put
$\beta=a/c$. If $|\beta|<1$ then $\lambda$ can be an eigenvalue only when
$|z_{1}|,|z_{2}|<1$; if $|\beta|>1$, then a necessary condition is that
$|z_{1}|,|z_{2}|>1$. Finaly, in the case when $|\beta|=1$, $\lambda$ is not an
eigenvalue unless $|z_{1}|=|z_{2}|=1$.

We discuss in more details the case when $|\beta|<1$. Then $z_{2}=\beta/z_{1}$
and the equation in Corollary 3.2 becomes

(11) $z\theta(z)=\frac{\beta}{z}\theta\Big{(}\frac{\beta}{z}\Big{)}.$

The question is whether there always exist solutions of this equation (apart
from the trivial one $z^{2}=\beta$) and how many such solutions can we have.
Consider the function

(12)
$\Psi(z)=\Big{(}z\theta(z)-\frac{\beta}{z}\theta\Big{(}\frac{\beta}{z}\Big{)}\Big{)}^{-1}.$

Since $\theta$ has a meromorphic pseudocontinuation to $\\{|z|>1\\}$ with
$|\theta(z)|>1$, we see that $\Psi$ is analytic and bounded in
$\\{|z|<|\beta|\\}$ and in $\\{|z|>1\\}$. Also, if the equation (11) has only
a finite number of solutions in the annulus $R_{\beta}=\\{|\beta|<|z|<1\\}$,
then $\Psi$ has a finite number of poles in this annulus. Note also that
$\Psi$ has bounded nontangential boundary values on $\\{|z|=|\beta|\\}$ and
$\\{|z|=1\\}$. One would like to show that in this case $\Psi$ is a rational
function which is impossible unless $\theta$ is a finite Blaschke product.
However, to “glue” the values of $\Psi$ in $R_{\beta}$ with the values in
$\\{|z|<|\beta|\\}$ and in $\\{|z|>1\\}$ one needs to pass to the boundary
limits. This is possible if we assume that $\Psi$ is of Smirnov class in
$R_{\beta}$ (for the theory of Hardy spaces in general domains see, e.g., [6,
Chapter 10]).

######  Theorem 3.3.

Let $\Phi(z)=a\bar{z}+b+cz$, $a,c\neq 0$, let $|\beta|=|c/a|<1$ and assume
that $\Psi$ is in the Smirnov class in the annulus $R_{\beta}$. If $\theta$ is
not a finite Blaschke product, then the equation (11) has infinitely many
solutions. In particular, $A_{\Phi}$ has infinitely many eigenvalues.

###### Proof.

Let $P$ be a polynomial with the zeros of corresponding multiplicities at the
poles of $\Psi$. Then the function $P\Psi$ is in the Smirnov class in
$R_{\beta}$ and has bounded nontangential boundary values on its boundary. By
the Smirnov theorem, $P\Psi$ is bounded in $R_{\beta}$. Also its nontangential
boundary values taken from inside of $R_{\beta}$ coincide with the boundary
values taken from the outer domains $\\{|z|<|\beta|\\}$ and $\\{|z|>1\\}$.
Then it is easy to see (taking the contours approaching the boundary from the
opposite sides and passing to the limit) that $P\Psi$ has an analytic
extension through the circles $\\{|z|=|\beta|\\}$ and $\\{|z|=1\\}$. Thus,
$P\Psi$ is an entire function of at most polynomial growth at infinity, whence
$\Psi$ is rational function. Since $\theta$ is not a finite Blaschke product,
this is impossible. ∎

In particular, if the boundary spectrum of $\theta$ is one point, e.g.,
$\sigma(\theta)=\\{1\\}$, then $\theta$ is meromorphic in
$\mathbb{C}\setminus\\{1\\}$, and the same is true for $\Psi$, if the equation
(11) has finite number of solutions. To show that $\Psi$ is in the Smirnov
class one needs only to show that $\Psi$ has no singular factor at $1$, that
is,

$\liminf_{r\to 1-}(1-r)\log|\Psi(r)|=0.$

######  Problem.

Is it true that the function
$\Psi(z)=\Big{(}z\theta(z)-\frac{\beta}{z}\theta\Big{(}\frac{\beta}{z}\Big{)}\Big{)}^{-1}$
is in the Smirnov class in $R_{\beta}=\\{|\beta|<|z|<1\\}$ for any inner
function $\theta$?

We believe that the answer to this question is positive, at least, for a wide
class of inner functions.

A result similar to Theorem 3.3 holds if $|\beta|=|c/a|>1$. Thus, we see that
under our assumptions on $\Psi$, the operators $\Phi(z)=a\bar{z}+b+cz$ with
$|a|\neq|c|$ have infinitely many eigenvalues. This leads to the following
corollary.

######  Corollary 3.4.

Let $\Phi(z)=a\bar{z}+b+cz$, $a,c\neq 0$, and $|a|\neq|c|$. Assume that for
any $\beta$ with $0<|\beta|<1$ the function $\Psi$ given by (12) is in the
Smirnov class in the annulus $R_{\beta}$. Then $A_{\Phi}$ is not hypercyclic.

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