Also, could I also issue notes based on the coins I hold? Or how would the bitcoin & Bitnote (quick invention
) coexist? Would it be the same to hold 1 BTC or the equivalent Bitnote? If the bank deflates the notes like banks did it with the gold backing, that could lead to value problems.
When the gold standard existed, dollars (and British pounds, French francs etc.) all existed alongside gold.
As for 'deflating', it all depends if you agree that money formation is endogenous or exogenous. Mainstream economics and monetarists assert that money is exogeneous - the supply is set and changed by central banks. In this scenario, deposits create loans: you put money in the bank they lend it out up to the limit they are allowed to and obtain reserves from the central bank. Central bank money is 'multiplied' through fractional reserve banking.
Keynesian, Post-Keynesian, Real-Business-Cycle and Marxian economists endorse endogenous money. Here, loans create deposits instead: a business seeks a loan to undertake a new project to expand their profits. They go to a bank who issues the loan
de novo - they basically create the money out of thin air. But that's fine- the company uses the loan money to pay their employees and their suppliers for this new project. The suppliers then pay their employees etc. This money in the form of wages and payments finds its way back to the bank (here our bank can represent the entire banking
system) in the form of deposits. If the reserves at the end of the day are not sufficient to support the loan demand, the bank will query the central bank who will only then print the money needed to supply the reserves.
This is a VERY nutshell version and the debate rages among economists as to which is the true description. But what is clear, is that it does not matter if the money is fiat or backed by a gold standard or whatever. The influence on the money (the notes, not the physical bullion in the case of the gold standard -- the gold is not money, the notes are) will be the same anyway.
For bitcoin, if bitnotes become a thing who can be the 'banks'? Miners can, to the degree that they obtain a steady flow of new money each day from 'the system'. But miners are competing with each other to such an extent that they lose profitability over time, so that might not work as well as it might seem at first. Those who have obtained a large supply of BTC through the open market or by conducting business in them could potentially create banknotes against their reserves, but if they are not actively obtaining more each day then they will go bust as soon as demand for notes exceeds their supply enough that a 'run' on the 'bank' would be ruinous. What about wallet services like coinbase or blockchain.info? They could create notes if they could issue loans denominated in BTC but that is not their business- they are payment and merchant services company. They are not interested in using Bitnotes as currency - they want bitcoin itself to be the good stuff. They could pay Bitcoin dividends to account holders from profits generated in their merchant services business but that would make account holders equity shareholders. This makes bitcoin an asset and not a money.
Bitnotes could earn interest as a debtor-creditor obligation and serve as money. But there is a problem in who can issue them without going bust given one or two bad loans.