Page 210 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 210
RECENT DEVELOPMENTS ON PRESERVERS (MS-38)
T : Mn(F) → Mn(F) preserving each of these relations. However, some additional assumptions
are needed, namely:
• for all relations (R, L, J , H, D), the classification was carried out under the assumption
that T is bijective;
• for the relations R, L, H, it was done under the assumption that the field F contains roots
of all polynomials from F[x] of degree n (particularly, for algebraically closed field);
• for the relations J , D, no additional assumptions are needed.
Also, for the relation H, it holds that non-zero H-preservers coincide exactly with invertibility
preservers, which classification is a well-known preserver problem. Over an arbitrary field,
linear maps preserving invertibility have been fully described by de Seguins Pazzis [4].
Furthermore, we show some examples of non-bijective linear L-, R-, and H-preservers that
are not of the form mentioned in the classification. This shows the importance of the restrictions
on the field or map stated above.
References
[1] J. A. Green. On the structure of semigroups. Annals of Math. 54 (1951), 163-172.
[2] G. Frobenius. Über die Darstellung der endlichen Gruppen durch lineare Substitutionen,
Sitzungsber. Deutsch. Akad. Wiss. (1897), 994-1015.
[3] A. Guterman, M. Johnson, M. Kambites. Linear isomorphisms preserving Green’s rela-
tions for matrices over anti-negative semifields, Linear Algebra Appl. 545 (2018), 1-14.
[4] C. de Seguins Pazzis. The singular linear preservers of non-singular matrices, Linear Al-
gebra Appl. 433 (2010), 483–490.
[5] A. Guterman, M. Johnson, M. Kambites, A. Maksaev. Linear functions preserving Green’s
relations over fields, Linear Algebra Appl. 611 (2021), 310-333.
The Gleason–Kahane–Z˙ elazko theorem and automatic countinuity
Javad Mashreghi, javad.mashreghi@mat.ulaval.ca
Laval University, Canada
The study of outer preserving linear maps on hardy spaces Hp led us to a generalized version of
Gleason–Kahane–Z˙ elazko theorem for modules. In particular, linear functionals T : Hp → C
(no continuity assumption) whose kernels do not include any outer function are not frequent
and should be of a very specific form which entails to automatic continuity. In this new work,
we take one step further and study such results in the general framework of reproducing kernel
Hilbert spaces. In a sense, this is the most general setting which include numerous special
function cases as a special case, e.g., Bergman, Dirichlet, Besov, the little Bloch, and VMOA.
208
T : Mn(F) → Mn(F) preserving each of these relations. However, some additional assumptions
are needed, namely:
• for all relations (R, L, J , H, D), the classification was carried out under the assumption
that T is bijective;
• for the relations R, L, H, it was done under the assumption that the field F contains roots
of all polynomials from F[x] of degree n (particularly, for algebraically closed field);
• for the relations J , D, no additional assumptions are needed.
Also, for the relation H, it holds that non-zero H-preservers coincide exactly with invertibility
preservers, which classification is a well-known preserver problem. Over an arbitrary field,
linear maps preserving invertibility have been fully described by de Seguins Pazzis [4].
Furthermore, we show some examples of non-bijective linear L-, R-, and H-preservers that
are not of the form mentioned in the classification. This shows the importance of the restrictions
on the field or map stated above.
References
[1] J. A. Green. On the structure of semigroups. Annals of Math. 54 (1951), 163-172.
[2] G. Frobenius. Über die Darstellung der endlichen Gruppen durch lineare Substitutionen,
Sitzungsber. Deutsch. Akad. Wiss. (1897), 994-1015.
[3] A. Guterman, M. Johnson, M. Kambites. Linear isomorphisms preserving Green’s rela-
tions for matrices over anti-negative semifields, Linear Algebra Appl. 545 (2018), 1-14.
[4] C. de Seguins Pazzis. The singular linear preservers of non-singular matrices, Linear Al-
gebra Appl. 433 (2010), 483–490.
[5] A. Guterman, M. Johnson, M. Kambites, A. Maksaev. Linear functions preserving Green’s
relations over fields, Linear Algebra Appl. 611 (2021), 310-333.
The Gleason–Kahane–Z˙ elazko theorem and automatic countinuity
Javad Mashreghi, javad.mashreghi@mat.ulaval.ca
Laval University, Canada
The study of outer preserving linear maps on hardy spaces Hp led us to a generalized version of
Gleason–Kahane–Z˙ elazko theorem for modules. In particular, linear functionals T : Hp → C
(no continuity assumption) whose kernels do not include any outer function are not frequent
and should be of a very specific form which entails to automatic continuity. In this new work,
we take one step further and study such results in the general framework of reproducing kernel
Hilbert spaces. In a sense, this is the most general setting which include numerous special
function cases as a special case, e.g., Bergman, Dirichlet, Besov, the little Bloch, and VMOA.
208
