In this paper two methods for generating continuous K-frames for a Hilbert space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal H$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> are introduced, where K is a bounded operator on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal H$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. Both methods transform a continuous K-frame of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal H$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> into another continuous K-frame of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal H$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. The first one has an algebraic approach: a continuous K-frame for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal H_A$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation> is shown to be preserved by some operators with specific algebraic properties, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal H_A$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation> denotes a module whose module operation depends on another fixed bounded operator A on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal H$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. The other method preserves a continuous K-frame using some minimal projections defined by means of the Moore-Penrose inverse of a closed range operator. Also, some examples illustrate the results.

Some linear maps preserving continuous frames for operators in Hilbert spaces

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发表日期：

2026年1月6日

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Salah Eddine Oustani,
Giorgia Bellomonte

摘要

In this paper two methods for generating continuous K-frames for a Hilbert space $\mathcal H$ $H$ are introduced, where K is a bounded operator on $\mathcal H$ $H$ . Both methods transform a continuous K-frame of $\mathcal H$ $H$ into another continuous K-frame of $\mathcal H$ $H$ . The first one has an algebraic approach: a continuous K-frame for $\mathcal H_A$ $H_{A}$ is shown to be preserved by some operators with specific algebraic properties, where $\mathcal H_A$ $H_{A}$ denotes a module whose module operation depends on another fixed bounded operator A on $\mathcal H$ $H$ . The other method preserves a continuous K-frame using some minimal projections defined by means of the Moore-Penrose inverse of a closed range operator. Also, some examples illustrate the results.