Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathscr {B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the algebra of all bounded linear operators on a complex Hilbert space H, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${{\mathscr {C}}}{{\mathscr {R}}}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the set of all operators in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\mathscr {B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with closed range. For any operator <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$A\in {\mathscr {B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$A^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$A^{\dag }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mo>†</mo> </msup> </math></EquationSource> </InlineEquation> be the adjoint and Moore–Penrose inverse of A, respectively. Fix a positive scalar <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Theta _\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> be the transformation defined on <InlineEquation ID="IEq12"> <EquationSource Format="TEX">${{\mathscr {C}}}{{\mathscr {R}}}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> by <Equation ID="Equ38"> <EquationSource Format="TEX">$\begin{aligned} \Theta _\alpha (A):= \frac{1}{\alpha +1}\left( \alpha A + A^{\dag }\right) ,~(A\in {{\mathscr {C}}}{{\mathscr {R}}}(H)). \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mfenced close=")" open="("> <mi>α</mi> <mi>A</mi> <mo>+</mo> <mmultiscripts> <mi>A</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> <mo>†</mo> </mrow> </mmultiscripts> </mfenced> <mo>,</mo> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∈</mo> <mi mathvariant="script">C</mi> <mi mathvariant="script">R</mi> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In this paper, we characterize all maps <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq14"> <EquationSource Format="TEX">${\mathscr {B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> preserving the closedness of the ranges of the difference of operators and for which <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\Theta _\alpha (\Phi (A)-\Phi (B))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\Theta _\alpha \left( A-B\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> <mfenced close=")" open="("> <mi>A</mi> <mo>-</mo> <mi>B</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> are equivalent by unitaries for all <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$A,~B\in {\mathscr {B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mspace width="3.33333pt" /> <mi>B</mi> <mo>∈</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Then we use such a characterization to obtain the form of all bijective linear maps <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq19"> <EquationSource Format="TEX">${\mathscr {B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for which <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\Theta _\alpha \left( \Phi (A)\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> <mfenced close=")" open="("> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\Theta _\alpha \left( \Phi (B)\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Θ</mi> <mi>α</mi> </msub> <mfenced close=")" open="("> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mfenced> </mrow> </math></EquationSource> </InlineEquation> are equivalent by unitaries whenever so are <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$A,~B\in {\mathscr {B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mspace width="3.33333pt" /> <mi>B</mi> <mo>∈</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we obtain similar results when the relation equivalence by unitaries is replaced by unitarily similarity. Furthermore, a number of related results and consequences is obtained.

Preservers of a New Transformation on ${\mathscr {B}}(H)$

期刊论文
发表日期：

2025年12月3日

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Abdellatif Bourhim,
Mostafa Mbekhta

摘要

Let ${\mathscr {B}}(H)$ $B (H)$ be the algebra of all bounded linear operators on a complex Hilbert space H, and ${{\mathscr {C}}}{{\mathscr {R}}}(H)$ $C R (H)$ be the set of all operators in ${\mathscr {B}}(H)$ $B (H)$ with closed range. For any operator $A\in {\mathscr {B}}(H)$ $A \in B (H)$ , let $A^*$ $A^{*}$ and $A^{\dag }$ $A^{†}$ be the adjoint and Moore–Penrose inverse of A, respectively. Fix a positive scalar $\alpha $ $α$ , and let $\Theta _\alpha $ $Θ_{α}$ be the transformation defined on ${{\mathscr {C}}}{{\mathscr {R}}}(H)$ $C R (H)$ by $\begin{aligned} \Theta _\alpha (A):= \frac{1}{\alpha +1}\left( \alpha A + A^{*\dag }\right) ,~(A\in {{\mathscr {C}}}{{\mathscr {R}}}(H)). \end{aligned}$ $\begin{matrix} Θ_{α} (A) : = \frac{1}{α + 1} (α A + A^{* †}), (A \in C R (H)) . \end{matrix}$ In this paper, we characterize all maps $\Phi $ $Φ$ on ${\mathscr {B}}(H)$ $B (H)$ preserving the closedness of the ranges of the difference of operators and for which $\Theta _\alpha (\Phi (A)-\Phi (B))$ $Θ_{α} (Φ (A) - Φ (B))$ and $\Theta _\alpha \left( A-B\right) $ $Θ_{α} (A - B)$ are equivalent by unitaries for all $A,~B\in {\mathscr {B}}(H)$ $A, B \in B (H)$ . Then we use such a characterization to obtain the form of all bijective linear maps $\Phi $ $Φ$ on ${\mathscr {B}}(H)$ $B (H)$ for which $\Theta _\alpha \left( \Phi (A)\right) $ $Θ_{α} (Φ (A))$ and $\Theta _\alpha \left( \Phi (B)\right) $ $Θ_{α} (Φ (B))$ are equivalent by unitaries whenever so are $A,~B\in {\mathscr {B}}(H)$ $A, B \in B (H)$ . Moreover, we obtain similar results when the relation equivalence by unitaries is replaced by unitarily similarity. Furthermore, a number of related results and consequences is obtained.

Preservers of a New Transformation on \({\mathscr {B}}(H)\)

摘要