Left Hilbert Algebras

Definition. Let H https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315273723/fa1df416-8e05-4351-832f-1e6189043d2c/content/eq4217.tif"/> and K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315273723/fa1df416-8e05-4351-832f-1e6189043d2c/content/eq4218.tif"/> be Hilbert spaces. A linear operator T from H https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315273723/fa1df416-8e05-4351-832f-1e6189043d2c/content/eq4219.tif"/> into K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315273723/fa1df416-8e05-4351-832f-1e6189043d2c/content/eq4220.tif"/> is said to be Hilbert–Schmidt if there is an orthonormal basis e _α of H https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315273723/fa1df416-8e05-4351-832f-1e6189043d2c/content/eq4221.tif"/> such that ∑ ║Te _α║² < ∞.