Author:
Mashreghi Javad,Ptak Marek,Ross William T.
Abstract
AbstractFor a unitary operator U on a separable complex Hilbert space $${\mathcal {H}}$$
H
, we describe the set $${\mathscr {C}}_{c}(U)$$
C
c
(
U
)
of all conjugations C (antilinear, isometric, and involutive maps) on $${\mathcal {H}}$$
H
for which $$C U C = U$$
C
U
C
=
U
. As this set might be empty, we also show that $${\mathscr {C}}_{c}(U) \not = \varnothing $$
C
c
(
U
)
≠
∅
if and only if U is unitarily equivalent to $$U^{*}$$
U
∗
.
Funder
Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada
Ministry of Science and Higher Education of the Republic of Poland
Publisher
Springer Science and Business Media LLC
Reference22 articles.
1. Bender, C.M.: Making sense of non-hermitian hamiltonians. Rep. Prog. Phys. 70(6), 947–1018 (2007)
2. Bender, C.M., Boettcher, S.: Real spectra in non-hermitian hamiltonians having $$\mathscr{P}\mathscr{T}$$ symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)
3. Cristina Câmara, M., Kliś-Garlicka, K., Ł anucha, B., Ptak, M.: Conjugations in $$L^2$$ and their invariants. Anal. Math. Phys. 10(2), 14 (2020)
4. Cristina Câmara, M., Kliś-Garlicka, K., Ł anucha, B., Ptak, M.: Conjugations in $$L^2(\cal{H} )$$. Integral Equ. Operator Theory 92(6), 25 (2020)
5. Conway, J.B.: A course in functional analysis. Graduate Texts in Mathematics, vol. 96. Springer-Verlag, New York (1985)