Paranormal operator

{{Short description|Operator type}}

In mathematics, especially operator theory, a paranormal operator is a generalization of a normal operator. More precisely, a bounded linear operator T on a complex Hilbert space H is said to be paranormal if:

: $\backslash |T^2x\backslash |\; \backslash ge\; \backslash |Tx\backslash |^2$

for every unit vector x in H.

The class of paranormal operators was introduced by V. Istratescu in 1960s, though the term "paranormal" is probably due to Furuta.{{cite journal

| last = Istrăţescu | first = V.

| journal = Pacific Journal of Mathematics

| mr = 213893

| pages = 413–417

| title = On some hyponormal operators

| url = https://projecteuclid.org/euclid.pjm/1102992095

| volume = 22

| year = 1967| issue = 3

| doi = 10.2140/pjm.1967.22.413

}}

Every hyponormal operator (in particular, a subnormal operator, a quasinormal operator and a normal operator) is paranormal. If T is a paranormal, then Tⁿ is paranormal.{{cite journal

| last = Furuta | first = Takayuki

| journal = Proceedings of the Japan Academy

| mr = 221302

| pages = 594–598

| title = On the class of paranormal operators

| url = https://projecteuclid.org/euclid.pja/1195521514

| volume = 43

| year = 1967}} On the other hand, Halmos gave an example of a hyponormal operator T such that T² isn't hyponormal. Consequently, not every paranormal operator is hyponormal.{{cite book

| last = Halmos | first = Paul Richard

| edition = 2nd

| isbn = 0-387-90685-1

| mr = 675952

| publisher = Springer-Verlag, New York-Berlin

| series = Encyclopedia of Mathematics and its Applications

| title = A Hilbert Space Problem Book

| volume = 17

| year = 1982}}

A compact paranormal operator is normal.{{cite journal

| last = Furuta | first = Takayuki

| journal = Proceedings of the Japan Academy

| mr = 313864

| pages = 888–893

| title = Certain convexoid operators

| url = https://projecteuclid.org/euclid.pja/1195526397

| volume = 47

| year = 1971| doi = 10.2183/pjab1945.47.SupplementI_888

}}