Frames for the Solution of Operator Equations in Hilbert Spaces with Fixed Dual Pairing

Balázs, Péter; Harbrecht, Helmut

doi:10.1080/01630563.2018.1495232

Cited by 8 publications

(8 citation statements)

References 52 publications

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“…Those concepts "only" deal with a triplet of spaces, while here we will work on the concept of frames and related objects on a full scale. Applications of frames for scales of Hilbert spaces can be found in the discretization of operators (as in [6,8,10,36,43]).…”

Section: Introductionmentioning

confidence: 99%

Frame-related Sequences in Chains and Scales of Hilbert Spaces

Balázs¹,

Bellomonte²,

Hosseinnezhad³

2022

Preprint

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Frames for Hilbert spaces are interesting for mathematicians but also important for applications e.g. in signal analysis and in physics. Both in mathematics and physics it is natural to consider a full scale of spaces, and not only a single one. In this paper, we study how certain frame-related properties, as completeness or the property of being a (semi-)frame, of a certain sequence in one of the spaces propagate to other spaces in a scale. We link that to the properties of the respective frame-related operators, like analysis or synthesis. We start with a detailed survey of the theory of Hilbert chains. Using a canonical isomorphism the properties of frame sequences are naturally preserved between different spaces. We also show that some results can be transferred if the original sequence is considered, in particular that the upper semi-frame property is kept in larger spaces, while the lower one to smaller ones. This leads to a negative result: a sequence can never be a frame for two Hilbert spaces of the scale if the scale is non-trivial, i.e. spaces are not equal.

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Section: Introductionmentioning

confidence: 99%

Frame-related Sequences in Chains and Scales of Hilbert Spaces

Balázs¹,

Bellomonte²,

Hosseinnezhad³

2022

Preprint

Self Cite

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“…Organic matter can act as a source of organic acids that can control the solubility of metals in the soil or act as a source of nutrients for plants. Organic acids found in organic matter are capable of igniting toxic substances in the soil so that they are not harmful to plants [6]. Organic acids could reduce the amount of phosphate fixed by Fe and Al through the mechanism of binding so that P is available to plants [7] [8].…”

Section: Introductionmentioning

confidence: 99%

Effectiveness of Organic Materials as Media in Sulfate Reducing Bacteria Inoculum to Changes on Acid Sulfate Soils

Mazlina

Hanafiah

Rauf

et al. 2021

Int. J. Eng. Scie. and Inform. Technology.

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Acid sulphate soils (ASS) had low pH, low nutrients availability and also soluble aluminium and iron were high. Inoculum of sulphate reducing bacteria (SRB) which organic matter as media could increased the soil pH, nutrient content and decrease sulfur-total of acid sulfate soils. The research was conducted in a randomized block design with two replications. This study used a randomized block design with two factors and two replications. The first factor was compost inoculum (C) was taken ten treatments from without any inoculum SRB and 9 treatment with different types and dosages of organic matter as media. The second factor was the water content condition (K) namely of K1: 100% field capacity and K2: 110% field capacity. Different types and dosages of organic matter appear to influence the changes in soil properties (tends to decrease soil sulfate and increase soil pH, and nutrient content levels in soil and plant). Inoculum SRB of palm oil empty bunches and weed gave a higher sulfate reduction compared to C0 (without inoculums) or inoculums with carrier media that used rice straw in water content 100% or 110% field capacity (FC).

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“…It is difficult to find a particular category of frames that is suitable to most of the physical problems, as there is no comprehensive class of frames that suits to all types of problems. Keeping this in mind, researchers across the disciplines come together for finding tools for the theory of frames to tackle various physical problems including solutions of operator equations in Hilbert spaces with fixed dual pairing [4]. Duffin and Schaeffer [12] defined frames as follows:…”

Section: Introductionmentioning

confidence: 99%

Trace class operators via OPV-frames

Bhardwaj

Sharma

Kaushik

2021

Filomat

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Trace class operators for quaternionic Hilbert spaces (QHS) were studied by Moretti and Oppio [18]. In this paper, we study trace class operators via operator valued frames (OPV-frames). We introduce OPV-frames in a right quaternionic Hilbert space H with range in a two sided quaternionic Hilbert space K and obtain various results including several characterizations of OPV-frames. Also, we obtain a necessary and sufficient condition for a bounded operator on a right QHS to be a trace class operator which generalizes a similar result by Attal [2]. Moreover, we construct a trace class operator on a two sided QHS. Finally, we study quaternionic quantum channels as completely positive trace preserving maps and obtain various Choi-Kraus type representations of quaternionic quantum channels using OPV-frames in quaternionic Hilbert spaces.

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Frames for the Solution of Operator Equations in Hilbert Spaces with Fixed Dual Pairing

Cited by 8 publications

References 52 publications

Frame-related Sequences in Chains and Scales of Hilbert Spaces

Frame-related Sequences in Chains and Scales of Hilbert Spaces

Effectiveness of Organic Materials as Media in Sulfate Reducing Bacteria Inoculum to Changes on Acid Sulfate Soils

Trace class operators via OPV-frames

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