A question related to spectral decomposition

Suppose that $ (\mathcal{A},\Phi)$ is a semifinite von Neumann algebra with trace $ \Phi$ , $ B$ is a positive self adjoint operator affiliated with $ \mathcal{A}$ , $ X$ is a positive operator with $ \Phi(X)<\infty$ that commutes with $ B$ , and $ E_n$ ‘s are an increasing sequence of spectral projections of $ X$ with $ \Phi(E_n)<\infty$ and $ \lim E_n=s(X)$ , where $ s(X)$ denotes the support of $ X$ . Let $ $ BX=\int_0^\infty \lambda dP(\lambda)$ $ be the spectral resolution for $ BX$ . Then $ $ BXE_n=\int_0^\infty \lambda dP_n(\lambda)$ $ where $ I-P_n(\lambda)=(I-P(\lambda))E_n$ . Why is it true that $ $ \Phi(BXE_n)=\int_0^\infty \Phi(I-P_n(\lambda))d\lambda?$ $

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I have a question about domain transfer

this may seem a newbie question but this is confusing for me.
If anyone can explain this, I would be greatful.

I have this website (domain+hosting) with a company. registered on Sep 18, 2018 for one year.
The hosting package + domain is going to expire on Sep 17, 2019.
I want to keep this domain but not with this company.
So do I have to transfer this domain or renew with another company?
And when?
If I transfer this domain to new company, today on June 23, 2019.
Will it be…

I have a question about domain transfer

[ Renting & Real Estate ] Open Question : Construction loan to mortgage loan. Can I get locked rate?

Looking into getting a construction loan that we will transfer to mortgage. Can I get pre-approval for the loan and have a locked in rate for the mortgage? Or does the mortgage rate go based on when the house is completed and we transfer the loan to the mortgage?

Post Frequency and Promotion Question for Evergreen blog articles

Hi all!

I have a quick question regarding releasing content.

I have written eight in-depth evergreen guides that are between 4000 and 5000 words.
The guides that I have created are much more readable and detailed than many of my competitor’s guides on the first page.
After reading through their guides, I noticed that many of their guides are obviously not written by native English speakers and lack proper punctuation and grammar. On the other hand, the blogs are easy to follow, contain relevant pictures and videos, and are appropriately formatted with keyword rich subheadings that were adequately researched, so I’m relatively confident that my blogs will rank well and generate traffic. I have applied this strategy to my product pages, and I am
ranking on the first page for quite a few terms, for the past few months. Now I am ready to take it to the next level and use informational search queries to convert curious shoppers into customers.

In addition to this, I have read that Google prefers sites that are updated regularly.

So my questions are:

Has anyone observed adverse effects if significant changes to sites all at once are preferred by Google or slow steady changes, in terms of weekly content distribution?

Should I get all of this information on my site right away or release it on a controlled schedule since Google prefers that publishers follow a predictable routine when releasing new work? I assume this helps with scheduling bots to crawl pages since Google has a limited budget to index the trillions of pages on the web.

Lastly, I intend to promote my blog posts via my social media channels, mailing lists, and affiliates. Are there any other services that I should be aware of to help promote my blog?

I work in the vape industry, so boosting posts on Facebook and Instagram may be out of the question unless they make exemptions or informational articles.

It’s competitive in the niche any advice is much appreciated. I am looking forward to hearing the experience of more seasoned SEO’s.

Thanks in advance for your time,